| 0000040490 |
a^2
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| 0000999900 |
b/(2a)
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| 0001030901 |
\cos(x)
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| 0001111111 |
(\sin(x))^2
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| 0001209482 |
2 \pi
|
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| 0001304952 |
\hbar
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| 0001334112 |
W
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| 0001921933 |
2 i
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| 0002239424 |
2
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| 0002338514 |
\vec{p}_{2}
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| 0002342425 |
m/m
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| 0002367209 |
1/2
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| 0002393922 |
x
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| 0002424922 |
a
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| 0002436656 |
i \hbar
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| 0002449291 |
b/(2a)
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| 0002838490 |
b/(2a)
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| 0002919191 |
\sin(-x)
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| 0002929944 |
1/2
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| 0002940021 |
2 \pi
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| 0003232242 |
t
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| 0003413423 |
\cos(-x)
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| 0003747849 |
-1
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| 0003838111 |
2
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| 0003919391 |
x
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| 0003949052 |
-x
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| 0003949921 |
\hbar
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| 0003954314 |
dx
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| 0003981813 |
-\sin(x)
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| 0004089571 |
2x
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| 0004264724 |
y
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| 0004307451 |
(b/(2a))^2
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| 0004582412 |
x
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| 0004829194 |
2
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| 0004831494 |
a
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| 0004849392 |
x
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| 0004858592 |
h
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| 0004934845 |
x
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| 0004948585 |
a
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| 0005395034 |
a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
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| 0005626421 |
t
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| 0005749291 |
f
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| 0005938585 |
\frac{-\hbar^2}{2m}
|
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| 0006466214 |
(\sin(x))^2
|
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| 0006544644 |
t
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| 0006563727 |
x
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| 0006644853 |
c/a
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| 0006656532 |
e
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| 0007471778 |
2(\sin(x))^2
|
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| 0007563791 |
i
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| 0007636749 |
x
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| 0007894942 |
(\sin(x))^2
|
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| 0008837284 |
T
|
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| 0008842811 |
\cos(2x)
|
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| 0009458842 |
\psi(x)
|
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| 0009484724 |
\frac{n \pi}{W}x
|
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| 0009485857 |
a^2\frac{2}{W}
|
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|
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| 0009485858 |
\frac{2n\pi}{W}
|
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| 0009492929 |
v du
|
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| 0009587738 |
\psi
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| 0009594995 |
1/2
|
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| 0009877781 |
y
|
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|
| 0203024440 |
1 = \int_0^W a \sin(\frac{n \pi}{W} x) \psi(x)^* dx
|
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|
|
|
| 0404050504 |
\lambda = \frac{v}{f}
|
|
|
|
|
| 0439492440 |
\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)\right|_0^W
|
|
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|
|
| 0934990943 |
k = \frac{2 \pi}{v T}
|
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|
|
| 0948572140 |
\int \cos(a x) dx = \frac{1}{a}\sin(a x)
|
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|
|
| 1010393913 |
\langle \psi| \hat{A}^+ |\psi \rangle = \langle a \rangle^*
|
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|
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| 1010393944 |
x = \langle\psi_{\alpha}| a_{\beta} |\psi_{\beta} \rangle
|
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|
|
| 1010923823 |
k W = n \pi
|
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|
|
| 1020010291 |
0 = a \sin(k W)
|
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|
|
| 1020394900 |
p = h/\lambda
|
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|
| 1020394902 |
E = h f
|
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|
|
| 1029039903 |
p = m v
|
|
|
|
|
| 1029039904 |
p^2 = m^2 v^2
|
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|
|
| 1158485859 |
\frac{-\hbar^2}{2m} \nabla^2 = {\cal H}
|
|
|
|
|
| 1202310110 |
\frac{1}{a^2} = \int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
|
|
|
|
|
| 1202312210 |
\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
|
|
|
|
|
| 1203938249 |
a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle = a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
|
|
|
|
|
| 1248277773 |
\cos(2x) = 1-2(\sin(x))^2
|
|
|
|
|
| 1293913110 |
0 = b
|
|
|
|
|
| 1293923844 |
\lambda = v T
|
|
|
|
|
| 1314464131 |
\vec{\nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
|
|
|
|
|
| 1314864131 |
\vec{\nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E}
|
|
|
|
|
| 1395858355 |
x = \langle \psi_{\alpha}| a_{\alpha} |\psi_{\beta}\rangle
|
|
|
|
|
| 1492811142 |
f = x - d
|
|
|
|
|
| 1492842000 |
\nabla \vec{x} = f(y)
|
|
|
|
|
| 1636453295 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = - \nabla^2 \vec{E}
|
|
|
|
|
| 1638282134 |
\vec{p}_{before} = \vec{p}_{after}
|
|
|
|
|
| 1648958381 |
\nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{\nabla}\psi(\vec{r},t)\right)
|
|
|
|
|
| 1857710291 |
0 = a \sin(n \pi)
|
|
|
|
|
| 1858578388 |
\nabla^2 E(\vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 1858772113 |
k = \frac{n \pi}{W}
|
|
|
|
|
| 1934748140 |
\int |\psi(x)|^2 dx = 1
|
|
|
|
|
| 2029293929 |
\nabla^2 E(\vec{r})\exp(i \omega t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 2103023049 |
\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right)
|
|
|
|
|
| 2113211456 |
f = 1/T
|
|
|
|
|
| 2123139121 |
-\exp(-i x) = -\cos(x)+i \sin(x)
|
|
|
|
|
| 2131616531 |
T f = 1
|
|
|
|
|
| 2258485859 |
{\cal H} \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
|
|
|
|
|
| 2394240499 |
x = a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle
|
|
|
|
|
| 2394853829 |
\exp(-i x) = \cos(-x)+i \sin(-x)
|
|
|
|
|
| 2394935831 |
( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
|
|
|
|
|
| 2394935835 |
\left(\langle\psi| \hat{A} |\psi\rangle \right)^+ = \left(\langle a \rangle\right)^+
|
|
|
|
|
| 2395958385 |
\nabla^2 \psi\left(\vec{r},t)\right) = \frac{-p^2}{\hbar} \psi(\vec{r},t)
|
|
|
|
|
| 2404934990 |
\langle x^2\rangle -2\langle x \rangle\langle x \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 2494533900 |
\langle x^2\rangle -\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 2569154141 |
\vec{\nabla} \times \frac{\partial}{\partial t}\vec{H} = \epsilon_0 \frac{\partial^2 }{\partial t^2}\vec{E}
|
|
|
|
|
| 2648958382 |
\nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p}\psi(\vec{r},t)\right)
|
|
|
|
|
| 2848934890 |
\langle a \rangle^* = \langle a \rangle
|
|
|
|
|
| 2944838499 |
\psi(x) = a \sin(\frac{n \pi}{W} x)
|
|
|
|
|
| 3121234211 |
\frac{k}{2\pi} = \lambda
|
|
|
|
|
| 3121234212 |
p = \frac{h k}{2\pi}
|
|
|
|
|
| 3121513111 |
k = \frac{2 \pi}{\lambda}
|
|
|
|
|
| 3131111133 |
T = 1 / f
|
|
|
|
|
| 3131211131 |
\omega = 2 \pi f
|
|
|
|
|
| 3132131132 |
\omega = \frac{2\pi}{T}
|
|
|
|
|
| 3147472131 |
\frac{\omega}{2 \pi} = f
|
|
|
|
|
| 3285732911 |
(\cos(x))^2 = 1-(\sin(x))^2
|
|
|
|
|
| 3485475729 |
\nabla^2 E(\vec{r}) = - \frac{\omega^2}{c^2} E(\vec{r})
|
|
|
|
|
| 3585845894 |
\langle \left(x-\langle x \rangle\right)^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 3829492824 |
\frac{1}{2}\left(\exp(i x)+\exp(-i x)\right) = \cos(x)
|
|
|
|
|
| 3924948349 |
a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
|
|
|
|
|
| 3942849294 |
\exp(i x)-\exp(-i x) = 2 i \sin(x)
|
|
|
|
|
| 3943939590 |
x = a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle
|
|
|
|
|
| 3947269979 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
|
|
|
|
|
| 3948571256 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi(\vec{r},t)
|
|
|
|
|
| 3948572230 |
\vec{\nabla}\psi(\vec{r},t) = \psi_0 \vec{\nabla}\exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
|
|
|
|
|
| 3948574224 |
\psi(\vec{r},t) = \psi_0 \exp\left(i\left(\vec{k}\cdot\vec{r} - \omega t \right) \right)
|
|
|
|
|
| 3948574226 |
\psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right)
|
|
|
|
|
| 3948574228 |
\psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
|
|
|
|
|
| 3948574230 |
\psi(\vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
|
|
|
|
|
| 3948574233 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
|
|
|
|
|
| 3948574235 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
|
|
|
|
|
| 3951205425 |
\vec{p}_{after} = \vec{p}_{1}
|
|
|
|
|
| 4147472132 |
E = \frac{h \omega}{2 \pi}
|
|
|
|
|
| 4298359835 |
E = \frac{1}{2}m v^2
|
|
|
|
|
| 4298359845 |
E = \frac{1}{2m}m^2 v^2
|
|
|
|
|
| 4298359851 |
E = \frac{p^2}{2m}
|
|
|
|
|
| 4341171256 |
i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{p^2}{2 m} \psi(\vec{r},t)
|
|
|
|
|
| 4348571256 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi(\vec{r},t)
|
|
|
|
|
| 4394958389 |
\vec{\nabla}\cdot \left(\vec{\nabla}\psi(\vec{r},t)\right) = \frac{i}{\hbar} \vec{\nabla}\cdot\left( \vec{p} \psi(\vec{r},t)\right)
|
|
|
|
|
| 4585828572 |
\epsilon_0 \mu_0 = \frac{1}{c^2}
|
|
|
|
|
| 4585932229 |
\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
|
|
|
|
|
| 4598294821 |
\exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
|
|
|
|
|
| 4638429483 |
\exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))
|
|
|
|
|
| 4742644828 |
\exp(i x)+\exp(-i x) = 2 \cos(x)
|
|
|
|
|
| 4827492911 |
\cos(2x)+(\sin(x))^2 = 1-(\sin(x))^2
|
|
|
|
|
| 4838429483 |
\exp(2 i x) = \cos(2 x)+i \sin(2 x)
|
|
|
|
|
| 4843995999 |
\frac{1}{2 i}\left(\exp(i x)-\exp(-i x)\right) = \sin(x)
|
|
|
|
|
| 4857472413 |
1 = \int \psi(x)\psi(x)^* dx
|
|
|
|
|
| 4857475848 |
\frac{1}{a^2} = \frac{W}{2}
|
|
|
|
|
| 4858429483 |
\exp(i x)\exp(i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))
|
|
|
|
|
| 4923339482 |
i x = \log(y)
|
|
|
|
|
| 4928239482 |
\log(y) = i x
|
|
|
|
|
| 4928923942 |
a = b
|
|
|
|
|
| 4929218492 |
a+b = c
|
|
|
|
|
| 4929482992 |
b = 2
|
|
|
|
|
| 4938429482 |
\exp(-i x) = \cos(x)+i \sin(-x)
|
|
|
|
|
| 4938429483 |
\exp(i x) = \cos(x)+i \sin(x)
|
|
|
|
|
| 4938429484 |
\exp(-i x) = \cos(x)-i \sin(x)
|
|
|
|
|
| 4943571230 |
\vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
|
|
|
|
|
| 4948934890 |
\langle \psi| \hat{A} |\psi\rangle = \langle a \rangle^*
|
|
|
|
|
| 4949359835 |
\langle x^2\rangle -2\langle x^2 \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 4954839242 |
\cos(2x)+i\sin(2x) = (\cos(x)+i \sin(x))(\cos(x)+i \sin(x))
|
|
|
|
|
| 4978429483 |
\exp(-i x) = \cos(-x)+i \sin(-x)
|
|
|
|
|
| 4984892984 |
a+2 = c
|
|
|
|
|
| 4985825552 |
\nabla^2 E(\vec{r})\exp(i \omega t) = i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 5530148480 |
\vec{p}_{1}-\vec{p}_{2} = \vec{p}_{electron}
|
|
|
|
|
| 5727578862 |
\frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)
|
|
|
|
|
| 5832984291 |
(\sin(x))^2 + (\cos(x))^2 = 1
|
|
|
|
|
| 5857434758 |
\int a dx = a x
|
|
|
|
|
| 5868688585 |
\frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = \frac{p^2}{2m} \psi(\vec{r},t)
|
|
|
|
|
| 5900595848 |
k = \frac{\omega}{v}
|
|
|
|
|
| 5928285821 |
x^2 + 2x(b/(2a)) + (b/(2a))^2 = (x+(b/(2a)))^2
|
|
|
|
|
| 5928292841 |
x^2 + (b/a)x + (b/(2a))^2 = -c/a + (b/(2a))^2
|
|
|
|
|
| 5938459282 |
x^2 + (b/a)x = -c/a
|
|
|
|
|
| 5958392859 |
x^2 + (b/a)x+(c/a) = 0
|
|
|
|
|
| 5959282914 |
x^2 + x(b/a) + (b/(2a))^2 = (x+(b/(2a)))^2
|
|
|
|
|
| 5982958248 |
x = -\sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
|
|
|
|
|
| 5982958249 |
x+(b/(2a)) = -\sqrt{(b/(2a))^2 - (c/a)}
|
|
|
|
|
| 5985371230 |
\vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi(\vec{r},t)
|
|
|
|
|
| 6742123016 |
\vec{p}_{electron}\cdot\vec{p}_{electron} = (\vec{p}_{1}\cdot\vec{p}_{1})+(\vec{p}_{2}\cdot\vec{p}_{2})-2(\vec{p}_{1}\cdot\vec{p}_{2})
|
|
|
|
|
| 7466829492 |
\vec{\nabla} \cdot \vec{E} = 0
|
|
|
|
|
| 7564894985 |
\int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)
|
|
|
|
|
| 7572664728 |
\cos(2x)+2(\sin(x))^2 = 1
|
|
|
|
|
| 7575738420 |
\left(\sin\left(\frac{n \pi}{W}x\right)\right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2}
|
|
|
|
|
| 7575859295 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859300 |
\epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{\nabla} \times \vec{E} )_k = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859302 |
\epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859304 |
\epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h}
|
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|
|
| 7575859306 |
\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859308 |
\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right) = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859310 |
\hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859312 |
\vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7917051060 |
\vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2}
|
|
|
|
|
| 8139187332 |
\vec{p}_{1} = \vec{p}_{2}+\vec{p}_{electron}
|
|
|
|
|
| 8257621077 |
\vec{p}_{before} = \vec{p}_{1}
|
|
|
|
|
| 8311458118 |
\vec{p}_{after} = \vec{p}_{2}+\vec{p}_{electron}
|
|
|
|
|
| 8399484849 |
\langle x^2 -2x\langle x \rangle+\langle x \rangle^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 8484544728 |
-a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x)
|
|
|
|
|
| 8485757728 |
a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(kx)
|
|
|
|
|
| 8485867742 |
\frac{2}{W} = a^2
|
|
|
|
|
| 8489593958 |
d(u v) = u dv + v du
|
|
|
|
|
| 8489593960 |
d(u v) - v du = u dv
|
|
|
|
|
| 8489593962 |
u dv = d(u v) - v du
|
|
|
|
|
| 8489593964 |
\int u dv = u v - \int v du
|
|
|
|
|
| 8494839423 |
\nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
|
|
|
|
|
| 8572657110 |
1 = \int |\psi(x)|^2 dx
|
|
|
|
|
| 8572852424 |
\vec{E} = E(\vec{r},t)
|
|
|
|
|
| 8575746378 |
\int \frac{1}{2} dx = \frac{1}{2} x
|
|
|
|
|
| 8575748999 |
\frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x)\right) = -k^2 \left(a \sin(kx) + b \cos(kx)\right)
|
|
|
|
|
| 8576785890 |
1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
|
|
|
|
|
| 8577275751 |
0 = a \sin(0) + b\cos(0)
|
|
|
|
|
| 8582885111 |
\psi(x) = a \sin(kx) + b \cos(kx)
|
|
|
|
|
| 8582954722 |
x^2 + 2xh + h^2 = (x+h)^2
|
|
|
|
|
| 8849289982 |
\psi(x)^* = a \sin(\frac{n \pi}{W} x)
|
|
|
|
|
| 8889444440 |
1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right)\right)^2 dx
|
|
|
|
|
| 9059289981 |
\psi(x) = a \sin(k x)
|
|
|
|
|
| 9285928292 |
ax^2 + bx + c = 0
|
|
|
|
|
| 9291999979 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0\vec{\nabla} \times \frac{\partial \vec{H}}{\partial t}
|
|
|
|
|
| 9294858532 |
\hat{A}^+ = \hat{A}
|
|
|
|
|
| 9385938295 |
(x+(b/(2a)))^2 = -(c/a) + (b/(2a))^2
|
|
|
|
|
| 9393939991 |
\psi(x) = -\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
|
|
|
|
|
| 9393939992 |
\psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
|
|
|
|
|
| 9394939493 |
\nabla^2 E(\vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r},t)
|
|
|
|
|
| 9429829482 |
\frac{d}{dx} y = -\sin(x) + i\cos(x)
|
|
|
|
|
| 9482113948 |
\frac{dy}{y} = i dx
|
|
|
|
|
| 9482438243 |
(\cos(x))^2 = \cos(2x)+(\sin(x))^2
|
|
|
|
|
| 9482923849 |
\exp(i x) = y
|
|
|
|
|
| 9482928242 |
\cos(2x) = (\cos(x))^2 - (\sin(x))^2
|
|
|
|
|
| 9482928243 |
\cos(2x)+(\sin(x))^2 = (\cos(x))^2
|
|
|
|
|
| 9482943948 |
\log(y) = i dx
|
|
|
|
|
| 9482948292 |
b = c
|
|
|
|
|
| 9482984922 |
\frac{d}{dx} y = (i\sin(x) + \cos(x)) i
|
|
|
|
|
| 9483928192 |
\cos(2x)+i\sin(2x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
|
|
|
|
|
| 9485384858 |
\nabla^2 E(\vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 9485747245 |
\sqrt{\frac{2}{W}} = a
|
|
|
|
|
| 9485747246 |
-\sqrt{\frac{2}{W}} = a
|
|
|
|
|
| 9492920340 |
y = \cos(x)+i \sin(x)
|
|
|
|
|
| 9494829190 |
k
|
|
|
|
|
| 9495857278 |
\psi(x=W) = 0
|
|
|
|
|
| 9499428242 |
E(\vec{r},t) = E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 9499959299 |
a + k = b + k
|
|
|
|
|
| 9582958293 |
x = \sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
|
|
|
|
|
| 9582958294 |
x+(b/(2a)) = \sqrt{(b/(2a))^2 - (c/a)}
|
|
|
|
|
| 9584821911 |
c + d = a
|
|
|
|
|
| 9585727710 |
\psi(x=0) = 0
|
|
|
|
|
| 9596004948 |
x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle
|
|
|
|
|
| 9848292229 |
dy = y i dx
|
|
|
|
|
| 9848294829 |
\frac{d}{dx} y = y i
|
|
|
|
|
| 9858028950 |
\frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
|
|
|
|
|
| 9889984281 |
2(\sin(x))^2 = 1-\cos(2x)
|
|
|
|
|
| 9919999981 |
\rho = 0
|
|
|
|
|
| 9958485859 |
\frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
|
|
|
|
|
| 9988949211 |
(\sin(x))^2 = \frac{1-\cos(2x)}{2}
|
|
|
|
|
| 9991999979 |
\vec{\nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t}
|
|
|
|
|
| 9999998870 |
\frac{\vec{p}}{\hbar} = \vec{k}
|
|
|
|
|
| 9999999870 |
\frac{p}{\hbar} = k
|
|
|
|
|
| 9999999950 |
\beta = 1/(k_{Boltzmann}T)
|
|
|
|
|
| 9999999951 |
\langle x | k \rangle = \frac{\exp(i k x)}{\sqrt{2\pi}}
|
|
|
|
|
| 9999999952 |
f(x) \delta(x-a) = f(a)
|
|
|
|
|
| 9999999953 |
\int_{-\infty}^{\infty} \delta(x) dx = 1
|
|
|
|
|
| 9999999954 |
c = 1/\sqrt{\epsilon_0 \mu_0}
|
|
|
|
|
| 9999999955 |
\vec{E} = -\vec{\nabla} \Psi
|
|
|
|
|
| 9999999956 |
\vec{F} = \frac{\partial}{\partial t}\vec{p}
|
|
|
|
|
| 9999999957 |
\vec{F} = -\vec{\nabla} V
|
|
|
|
|
| 9999999960 |
\hbar = h/(2 \pi)
|
|
|
|
|
| 9999999961 |
\frac{E}{\hbar} = \omega
|
|
|
|
|
| 9999999962 |
p = \hbar k
|
|
|
|
|
| 9999999963 |
\lambda = h/p
|
|
|
|
|
| 9999999964 |
\omega = c k
|
|
|
|
|
| 9999999965 |
E = \omega \hbar
|
|
|
|
|
| 9999999966 |
\vec{L} = \vec{r}\times\vec{p}
|
|
|
|
|
| 9999999967 |
\vec{S} = \frac{1}{\mu_0} \vec{E}\times \vec{B}
|
|
|
|
|
| 9999999968 |
x = \frac{-b-\sqrt{b^2-4ac}}{2a}
|
|
|
|
|
| 9999999969 |
x = \frac{-b+\sqrt{b^2-4ac}}{2a}
|
|
|
|
|
| 9999999970 |
\eta_1 \sin\theta_1 = \eta_2 \sin\theta_2
|
|
|
|
|
| 9999999971 |
{\cal H} = \frac{p^2}{2m}+V
|
|
|
|
|
| 9999999972 |
{\cal H} |\psi\rangle = E |\psi\rangle
|
|
|
|
|
| 9999999973 |
\left( \Delta A \right)^2 = \langle A^2 \rangle - \langle A \rangle^2
|
|
|
|
|
| 9999999974 |
\langle \psi| \hat{A} |\psi\rangle = \int_{-\infty}^{\infty} \psi^* A \psi dx
|
|
|
|
|
| 9999999975 |
\langle \psi| \hat{A} |\psi\rangle = \langle a \rangle
|
|
|
|
|
| 9999999976 |
\hat{p} = -i \hbar \frac{\partial }{\partial x}
|
|
|
|
|
| 9999999977 |
[\hat{x},\hat{p}] = i \hbar
|
|
|
|
|
| 9999999978 |
\vec{\nabla} \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial E}{\partial t}
|
|
|
|
|
| 9999999979 |
\vec{\nabla} \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}
|
|
|
|
|
| 9999999980 |
\vec{\nabla} \cdot \vec{B} = 0
|
|
|
|
|
| 9999999981 |
\vec{\nabla} \cdot \vec{E} = \rho/\epsilon_0
|
|
|
|
|
| 9999999982 |
V = I R + Q/C + L \frac{\partial I}{\partial t}
|
|
|
|
|
| 9999999983 |
C = V A/d
|
|
|
|
|
| 9999999984 |
Q = C V
|
|
|
|
|
| 9999999985 |
V = I R
|
|
|
|
|
| 9999999986 |
\left[\right]\left[\right] = \left[\right]
|
|
|
|
|
| 9999999987 |
1 atmosphere = 101.325 Pascal
|
|
|
|
|
| 9999999988 |
1 atmosphere = 14.7 pounds/(inch^2)
|
|
|
|
|
| 9999999989 |
mass_{electron} = 511000 electronVolts/(q^2)
|
|
|
|
|
| 9999999990 |
Tesla = 10000*Gauss
|
|
|
|
|
| 9999999991 |
Pascal = Newton / (meter^2)
|
|
|
|
|
| 9999999992 |
Tesla = Newton / (Ampere*meter)
|
|
|
|
|
| 9999999993 |
Farad = Coulumb / Volt
|
|
|
|
|
| 9999999995 |
Volt = Joule / Coulumb
|
|
|
|
|
| 9999999996 |
Coulomb = Ampere / second
|
|
|
|
|
| 9999999997 |
Watt = Joule / second
|
|
|
|
|
| 9999999998 |
Joule = Newton*meter
|
|
|
|
|
| 9999999999 |
Newton = kilogram*meter/(second^2)
|
|
|
|
|
| 4961662865 |
x
|
|
|
|
|
| 9110536742 |
2 x
|
|
|
|
|
| 8483686863 |
\sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x)\right)
|
|
|
|
|
| 3470587782 |
\sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
|
|
|
|
|
| 8642992037 |
2
|
|
|
|
|
| 9894826550 |
2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \left(\exp(i x)+\exp(-i x)\right)
|
|
|
|
|
| 4257214632 |
\frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)
|
|
|
|
|
| 8699789241 |
2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)
|
|
|
|
|
| 9180861128 |
2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)
|
|
|
|
|
| 2405307372 |
\sin(2 x) = 2 \sin(x) \cos(x)
|
|
|
|
|
| 3268645065 |
x
|
|
|
|
|
| 9350663581 |
\pi
|
|
|
|
|
| 8332931442 |
\exp(i \pi) = \cos(\pi)+i \sin(\pi)
|
|
|
|
|
| 6885625907 |
\exp(i \pi) = -1 + i 0
|
|
|
|
|
| 3331824625 |
\exp(i \pi) = -1
|
|
|
|
|
| 4901237716 |
1
|
|
|
|
|
| 2501591100 |
\exp(i \pi) + 1 = 0
|
|
|
|
|
| 5136652623 |
E = KE + PE
|
|
mechanical energy is the sum of the potential plus kinetic energies |
|
|
| 7915321076 |
E, KE, PE
|
|
|
|
|
| 3192374582 |
E_2, KE_2, PE_2
|
|
|
|
|
| 7875206161 |
E_2 = KE_2 + PE_2
|
|
|
|
|
| 4229092186 |
E, KE, PE
|
|
|
|
|
| 1490821948 |
E_1, KE_1, PE_1
|
|
|
|
|
| 4303372136 |
E_1 = KE_1 + PE_1
|
|
|
|
|
| 5514556106 |
E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1)
|
|
|
|
|
| 8357234146 |
KE = \frac{1}{2} m v^2
|
|
kinetic energy |
Kinetic_energy
|
|
| 3658066792 |
KE_2, v_2
|
|
|
|
|
| 8082557839 |
KE, v
|
|
|
|
|
| 7676652285 |
KE_2 = \frac{1}{2} m v_2^2
|
|
|
|
|
| 2790304597 |
KE_1, v_1
|
|
|
|
|
| 9880327189 |
KE, v
|
|
|
|
|
| 4928007622 |
KE_1 = \frac{1}{2} m v_1^2
|
|
|
|
|
| 5733146966 |
KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right)
|
|
|
|
|
| 6554292307 |
t
|
|
|
|
|
| 4270680309 |
\frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t}
|
|
|
|
|
| 5781981178 |
x^2 - y^2 = (x+y)(x-y)
|
|
difference of squares |
Difference_of_two_squares
|
|
| 5946759357 |
v_2, v_1
|
|
|
|
|
| 6675701064 |
x, y
|
|
|
|
|
| 4648451961 |
v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1)
|
|
|
|
|
| 9356924046 |
\frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t}
|
|
|
|
|
| 9397152918 |
v = \frac{v_1 + v_2}{2}
|
|
average velocity |
|
|
| 2857430695 |
a = \frac{v_2 - v_1}{t}
|
|
acceleration |
|
|
| 7735737409 |
\frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t}
|
|
|
|
|
| 4784793837 |
\frac{KE_2 - KE_1}{t} = m v a
|
|
|
|
|
| 5345738321 |
F = m a
|
|
Newton's second law of motion |
Newton%27s_laws_of_motion#Newton's_second_law
|
|
| 2186083170 |
\frac{KE_2 - KE_1}{t} = v F
|
|
|
|
|
| 6715248283 |
PE = -F x
|
|
potential energy |
Potential_energy
|
|
| 3852078149 |
PE_2, x_2
|
|
|
|
|
| 7388921188 |
PE, x
|
|
|
|
|
| 2431507955 |
PE_2 = -F x_2
|
|
|
|
|
| 3412641898 |
PE_1, x_1
|
|
|
|
|
| 9937169749 |
PE, x
|
|
|
|
|
| 4669290568 |
PE_1 = -F x_1
|
|
|
|
|
| 7734996511 |
PE_2 - PE_1 = -F ( x_2 - x_1 )
|
|
|
|
|
| 2016063530 |
t
|
|
|
|
|
| 7267155233 |
\frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right)
|
|
|
|
|
| 9337785146 |
v = \frac{x_2 - x_1}{t}
|
|
average velocity |
|
|
| 4872970974 |
\frac{PE_2 - PE_1}{t} = -F v
|
|
|
|
|
| 2081689540 |
t
|
|
|
|
|
| 2770069250 |
\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t}
|
|
|
|
|
| 3591237106 |
\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v
|
|
|
|
|
| 1772416655 |
\frac{E_2 - E_1}{t} = v F - F v
|
|
|
|
|
| 1809909100 |
\frac{E_2 - E_1}{t} = 0
|
|
|
|
|
| 5778176146 |
t
|
|
|
|
|
| 3806977900 |
E_2 - E_1 = 0
|
|
|
|
|
| 5960438249 |
E_1
|
|
|
|
|
| 8558338742 |
E_2 = E_1
|
|
|
|
|
| 8945218208 |
\theta_{Brewster} + \theta_{refracted} = 90^{\circ}
|
|
|
based on figure 34-27 on page 824 in 2001_HRW
|
|
| 9025853427 |
\theta_{Brewster}
|
|
|
|
|
| 1310571337 |
\theta_{refracted} = 90^{\circ} - \theta_{Brewster}
|
|
|
|
|
| 6450985774 |
n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 )
|
|
Law of Refraction |
eq 34-44 on page 819 in 2001_HRW
|
|
| 1779497048 |
\theta_{Brewster}, \theta_{refraction}
|
|
|
|
|
| 7322344455 |
\theta_1, \theta_2
|
|
|
|
|
| 2575937347 |
n_1 \sin( \theta_{Brewster} ) = n_2 \sin( \theta_{refraction} )
|
|
|
|
|
| 7696214507 |
n_1 \sin( \theta_{Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{Brewster} )
|
|
|
|
|
| 8588429722 |
\sin( 90^{\circ} - x ) = \cos( x )
|
|
|
|
|
| 7375348852 |
\theta_{Brewster}
|
|
|
|
|
| 1512581563 |
x
|
|
|
|
|
| 6831637424 |
\sin( 90^{\circ} - \theta_{Brewster} ) = \cos( \theta_{Brewster} )
|
|
|
|
|
| 3061811650 |
n_1 \sin( \theta_{Brewster} ) = n_2 \cos( \theta_{Brewster} )
|
|
|
|
|
| 7857757625 |
n_1
|
|
|
|
|
| 9756089533 |
\sin( \theta_{Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{Brewster} )
|
|
|
|
|
| 5632428182 |
\cos( \theta_{Brewster} )
|
|
|
|
|
| 2768857871 |
\frac{\sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )} = \frac{n_2}{n_1}
|
|
|
|
|
| 4968680693 |
\tan( x ) = \frac{ \sin( x )}{\cos( x )}
|
|
|
|
|
| 7321695558 |
\theta_{Brewster}
|
|
|
|
|
| 9906920183 |
x
|
|
|
|
|
| 4501377629 |
\tan( \theta_{Brewster} ) = \frac{ \sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )}
|
|
|
|
|
| 3417126140 |
\tan( \theta_{Brewster} ) = \frac{ n_2 }{ n_1 }
|
|
|
|
|
| 5453995431 |
\arctan{ x }
|
|
|
|
|
| 6023986360 |
x
|
|
|
|
|
| 8495187962 |
\theta_{Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }
|
|
|
|
|
| 9040079362 |
f
|
|
|
|
|
| 9565166889 |
T
|
|
|
|
|
| 5426308937 |
v = \frac{d}{t}
|
|
|
|
|
| 7476820482 |
C
|
|
|
|
|
| 1277713901 |
d
|
|
|
|
|
| 6946088325 |
v = \frac{C}{t}
|
|
|
|
|
| 6785303857 |
C = 2 \pi r
|
|
|
|
|
| 4816195267 |
C, r
|
|
|
|
|
| 2113447367 |
C_{\rm{Earth\ orbit}}, r_{\rm{Earth\ orbit}}
|
|
|
|
|
| 6348260313 |
C_{\rm{Earth\ orbit}} = 2 \pi r_{\rm{Earth\ orbit}}
|
|
|
|
|
| 3847777611 |
C, v, t
|
|
|
|
|
| 6406487188 |
C_{\rm{Earth\ orbit}}, v_{\rm{Earth\ orbit}}, t_{\rm{Earth\ orbit}}
|
|
|
|
|
| 3046191961 |
v_{\rm{Earth\ orbit}} = \frac{C_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
|
|
|
|
|
| 3080027960 |
v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
|
|
|
|
|
| 7175416299 |
t_{\rm{Earth\ orbit}} = 1 {\rm{year}}
|
|
|
|
|
| 3219318145 |
\frac{365 {\rm days}}{1 {\rm year}} \frac{24 {\rm hours}}{1 {\rm day}} \frac{60 {\rm minutes}}{1 {\rm hour}} \frac{60 {\rm seconds}}{1 {\rm minute}}
|
|
|
|
|
| 8721295221 |
t_{\rm{Earth\ orbit}} = 3.16 10^7 {\rm{seconds}}
|
|
|
|
|
| 4593428198 |
v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{3.16\ 10^7 {\rm seconds}}
|
|
|
|
|
| 3472836147 |
r_{\rm{Earth\ orbit}} = 1.496\ 10^8 {\rm km}
|
|
|
|
|
| 6998364753 |
v_{\rm{Earth\ orbit}} = \frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}}
|
|
|
|
|
| 4180845508 |
v_{\rm{Earth\ orbit}} = 29.8 \frac{{\rm km}}{{\rm sec}}
|
|
|
|
|
| 4662369843 |
x' = \gamma (x - v t)
|
|
|
|
|
| 2983053062 |
x = \gamma (x' + v t')
|
|
|
|
|
| 3426941928 |
x = \gamma ( \gamma (x - v t) + v t' )
|
|
|
|
|
| 2096918413 |
x = \gamma ( \gamma x - \gamma v t + v t' )
|
|
|
|
|
| 7741202861 |
x = \gamma^2 x - \gamma^2 v t + \gamma v t'
|
|
|
|
|
| 7337056406 |
\gamma^2 x
|
|
|
|
|
| 4139999399 |
x - \gamma^2 x = - \gamma^2 v t + \gamma v t'
|
|
|
|
|
| 3495403335 |
x
|
|
|
|
|
| 9031609275 |
x (1 - \gamma^2 ) = - \gamma^2 v t + \gamma v t'
|
|
|
|
|
| 8014566709 |
\gamma^2 v t
|
|
|
|
|
| 9409776983 |
x (1 - \gamma^2 ) + \gamma^2 v t = \gamma v t'
|
|
|
|
|
| 2226340358 |
\gamma v
|
|
|
|
|
| 6417705759 |
\frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = \gamma v t'
|
|
|
|
|
| 8730201316 |
\frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t = t'
|
|
|
first term was multiplied by \gamma/\gamma
|
|
| 1974334644 |
\frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = t'
|
|
|
|
|
| 5148266645 |
t' = \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t
|
|
|
|
|
| 4287102261 |
x^2 + y^2 + z^2 = c^2 t^2
|
|
|
describes a spherical wavefront
|
|
| 1201689765 |
x'^2 + y'^2 + z'^2 = c^2 t'^2
|
|
|
describes a spherical wavefront for an observer in a moving frame of reference
|
|
| 7057864873 |
y' = y
|
|
|
frame of reference is moving only along x direction
|
|
| 8515803375 |
z' = z
|
|
|
frame of reference is moving only along x direction
|
|
| 6153463771 |
3
|
|
|
|
|
| 4746576736 |
asdf
|
|
|
|
|
| 1027270623 |
minga
|
|
|
|
|
| 6101803533 |
ming
|
|
|
|
|
| 9231495607 |
a = b
|
|
|
|
|
| 8664264194 |
b + 2
|
|
|
|
|
| 9805063945 |
\gamma^2 (x - v t)^2 + y^2 + z^2 = c^2 \gamma^2 \left( t + \frac{ 1 - \gamma^2 }{ \gamma^2 } \frac{x}{v} \right)^2
|
|
|
|
|
| 4057686137 |
C \to C_{\rm{Earth\ orbit}}
|
|
|
|
|
| 2346150725 |
r \to r_{\rm{Earth\ orbit}}
|
|
|
|
|
| 9753878784 |
v \to v_{\rm{Earth\ orbit}}
|
|
|
|
|
| 8135396036 |
t \to t_{\rm{Earth\ orbit}}
|
|
|
|
|
| 2773628333 |
\theta_1 \to \theta_{Brewster}
|
|
|
|
|
| 7154592211 |
\theta_2 \to \theta_{\rm refracted}
|
|
|
|
|
| 3749492596 |
E \to E_1
|
|
|
|
|
| 1258245373 |
E \to E_2
|
|
|
|
|
| 4147101187 |
KE \to KE_1
|
|
|
|
|
| 3809726424 |
PE \to PE_1
|
|
|
|
|
| 6383056612 |
KE \to KE_2
|
|
|
|
|
| 5075406409 |
PE \to PE_2
|
|
|
|
|
| 5398681503 |
v \to v_1
|
|
|
|
|
| 9305761407 |
v \to v_2
|
|
|
|
|
| 4218009993 |
x \to x_1
|
|
|
|
|
| 4188639044 |
x \to x_2
|
|
|
|
|
| 8173074178 |
x \to v_2
|
|
|
|
|
| 1025759423 |
y \to v_1
|
|
|
|
|
| 1935543849 |
\gamma^2 x^2 - \gamma^2 2 x v t + \gamma^2 v^2 t^2 + y^2 + z^2 = c^2 \gamma^2 \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x^2}{\gamma^2} + c^2 \gamma^2 2 t \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x}{\gamma} + c^2 \gamma^2 t^2
|
|
|
|
|
| 1586866563 |
\left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right) = t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)
|
|
|
|
|
| 3182633789 |
\gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1
|
|
|
|
|
| 1916173354 |
-\gamma^2 v^2 + c^2 \gamma^2 = c^2
|
|
|
|
|
| 2076171250 |
-\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} = 0
|
|
|
|
|
| 5284610349 |
\gamma^2
|
|
|
|
|
| 2417941373 |
- c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 - \gamma^2
|
|
|
|
|
| 5787469164 |
1 - \gamma^2
|
|
|
|
|
| 1639827492 |
- c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2} = 1
|
|
|
|
|
| 5669500954 |
v^2 \gamma^2
|
|
|
|
|
| 5763749235 |
-c^2 + c^2 \gamma^2 = v^2 \gamma^2
|
|
|
|
|
| 6408214498 |
c^2
|
|
|
|
|
| 2999795755 |
c^2 \gamma^2 = v^2 \gamma^2 + c^2
|
|
|
|
|
| 3412946408 |
v^2 \gamma^2
|
|
|
|
|
| 2542420160 |
c^2 \gamma^2 - v^2 \gamma^2 = c^2
|
|
|
|
|
| 7743841045 |
\gamma^2
|
|
|
|
|
| 7513513483 |
\gamma^2 (c^2 - v^2) = c^2
|
|
|
|
|
| 8571466509 |
c^2 - \gamma^2
|
|
|
|
|
| 7906112355 |
\gamma^2 = \frac{c^2}{c^2 - \gamma^2}
|
|
|
|
|
| 3060139639 |
1/2
|
|
|
|
|
| 1528310784 |
\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
|
|
|
|
|
| 8360117126 |
\gamma = \frac{-1}{\sqrt{1-\frac{v^2}{c^2}}}
|
|
|
not a physically valid result in this context
|
|
| 3366703541 |
a = \frac{v - v_0}{t}
|
|
|
acceleration is the average change in speed over a duration
|
|
| 7083390553 |
t
|
|
|
|
|
| 4748157455 |
a t = v - v_0
|
|
|
|
|
| 6417359412 |
v_0
|
|
|
|
|
| 4798787814 |
a t + v_0 = v
|
|
|
|
|
| 3462972452 |
v = v_0 + a t
|
|
|
|
|
| 3411994811 |
v_{\rm ave} = \frac{d}{t}
|
|
|
|
|
| 6175547907 |
v_{\rm ave} = \frac{v + v_0}{2}
|
|
|
|
|
| 9897284307 |
\frac{d}{t} = \frac{v + v_0}{2}
|
|
|
|
|
| 8865085668 |
t
|
|
|
|
|
| 8706092970 |
d = \left(\frac{v + v_0}{2}\right)t
|
|
|
|
|
| 7011114072 |
d = \frac{(v_0 + a t) + v_0}{2} t
|
|
|
|
|
| 1265150401 |
d = \frac{2 v_0 + a t}{2} t
|
|
|
|
|
| 9658195023 |
d = v_0 t + \frac{1}{2} a t^2
|
|
|
|
|
| 5799753649 |
2
|
|
|
|
|
| 7215099603 |
v^2 = v_0^2 + 2 a t v_0 + a^2 t^2
|
|
|
|
|
| 5144263777 |
v^2 = v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right)
|
|
|
|
|
| 7939765107 |
v^2 = v_0^2 + 2 a d
|
|
|
|
|
| 6729698807 |
v_0
|
|
|
|
|
| 9759901995 |
v - v_0 = a t
|
|
|
|
|
| 2242144313 |
a
|
|
|
|
|
| 1967582749 |
t = \frac{v - v_0}{a}
|
|
|
|
|
| 5733721198 |
d = \frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right)
|
|
|
|
|
| 5611024898 |
d = \frac{1}{2 a} (v^2 - v_0^2)
|
|
|
|
|
| 5542390646 |
2 a
|
|
|
|
|
| 8269198922 |
2 a d = v^2 - v_0^2
|
|
|
|
|
| 9070454719 |
v_0^2
|
|
|
|
|
| 4948763856 |
2 a d + v_0^2 = v^2
|
|
|
|
|
| 9645178657 |
a t
|
|
|
|
|
| 6457044853 |
v - a t = v_0
|
|
|
|
|
| 1259826355 |
d = (v - a t) t + \frac{1}{2} a t^2
|
|
|
|
|
| 4580545876 |
d = v t - a t^2 + \frac{1}{2} a t^2
|
|
|
|
|
| 6421241247 |
d = v t - \frac{1}{2} a t^2
|
|
|
|
|
| 4182362050 |
Z = |Z| \exp( i \theta )
|
|
|
Z \in \mathbb{C}
|
|
| 1928085940 |
Z^* = |Z| \exp( -i \theta )
|
|
|
|
|
| 9191880568 |
Z Z^* = |Z| |Z| \exp( -i \theta ) \exp( i \theta )
|
|
|
|
|
| 3350830826 |
Z Z^* = |Z|^2
|
|
|
|
|
| 8396997949 |
I = | A + B |^2
|
|
|
intensity of two waves traveling opposite directions on same path
|
|
| 4437214608 |
Z
|
|
|
|
|
| 5623794884 |
A + B
|
|
|
|
|
| 2236639474 |
(A + B)(A + B)^* = |A + B|^2
|
|
|
|
|
| 1020854560 |
I = (A + B)(A + B)^*
|
|
|
|
|
| 6306552185 |
I = (A + B)(A^* + B^*)
|
|
|
|
|
| 8065128065 |
I = A A^* + B B^* + A B^* + B A^*
|
|
|
|
|
| 9761485403 |
Z
|
|
|
|
|
| 8710504862 |
A
|
|
|
|
|
| 4075539836 |
A A^* = |A|^2
|
|
|
|
|
| 6529120965 |
B
|
|
|
|
|
| 1511199318 |
Z
|
|
|
|
|
| 7107090465 |
B B^* = |B|^2
|
|
|
|
|
| 5125940051 |
I = |A|^2 + B B^* + A B^* + B A^*
|
|
|
|
|
| 1525861537 |
I = |A|^2 + |B|^2 + A B^* + B A^*
|
|
|
|
|
| 1742775076 |
Z \to B
|
|
|
|
|
| 7607271250 |
\theta \to \phi
|
|
|
|
|
| 4192519596 |
B = |B| \exp(i \phi)
|
|
|
|
|
| 2064205392 |
A
|
|
|
|
|
| 1894894315 |
Z
|
|
|
|
|
| 1357848476 |
A = |A| \exp(i \theta)
|
|
|
|
|
| 6555185548 |
A^* = |A| \exp(-i \theta)
|
|
|
|
|
| 4504256452 |
B^* = |B| \exp(-i \phi)
|
|
|
|
|
| 7621705408 |
I = |A|^2 + |B|^2 + |A| |B| \exp(-i \theta) \exp(i \phi) + |A| |B| \exp(i \theta) \exp(-i \phi)
|
|
|
|
|
| 3085575328 |
I = |A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi))
|
|
|
|
|
| 4935235303 |
x
|
|
|
|
|
| 2293352649 |
\theta - \phi
|
|
|
|
|
| 3660957533 |
\cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
|
|
|
|
|
| 3967985562 |
2
|
|
|
|
|
| 2700934933 |
2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
|
|
|
|
|
| 8497631728 |
I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )
|
|
|
|
|
| 6774684564 |
\theta = \phi
|
|
|
for coherent waves
|
|
| 2099546947 |
I = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )
|
|
|
|
|
| 2719691582 |
|A| = |B|
|
|
|
in a loop
|
|
| 3257276368 |
I = |A|^2 + |A|^2 + |A| |A| 2
|
|
|
|
|
| 8283354808 |
I_{\rm coherent} = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )
|
|
|
|
|
| 8046208134 |
I_{\rm coherent} = |A|^2 + |A|^2 + |A| |A| 2
|
|
|
|
|
| 1172039918 |
I_{\rm coherent} = 4 |A|^2
|
|
|
|
|
| 8602221482 |
\langle \cos(\theta - \phi) \rangle = 0
|
|
|
incoherent light source
|
|
| 6240206408 |
I_{\rm incoherent} = |A|^2 + |B|^2
|
|
|
|
|
| 6529793063 |
I_{\rm incoherent} = |A|^2 + |A|^2
|
|
|
|
|
| 3060393541 |
I_{\rm incoherent} = 2|A|^2
|
|
|
|
|
| 6556875579 |
\frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2
|
|
|
|
|
| 8750500372 |
f
|
|
|
|
|
| 7543102525 |
g
|
|
|
|
|
| 8267133829 |
a = b
|
|
|
|
|
| 7968822469 |
c = d
|
|
|
|
|
| 3169580383 |
\vec{a} = \frac{d\vec{v}}{dt}
|
|
|
acceleration is the change in speed over a duration
|
|
| 8602512487 |
\vec{a} = a_x \hat{x} + a_y \hat{y}
|
|
|
decompose acceleration into two components
|
|
| 4158986868 |
a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt}
|
|
|
|
|
| 5349866551 |
\vec{v} = v_x \hat{x} + v_y \hat{y}
|
|
|
|
|
| 7729413831 |
a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right)
|
|
|
|
|
| 1819663717 |
a_x = \frac{d}{dt} v_x
|
|
|
|
|
| 8228733125 |
a_y = \frac{d}{dt} v_y
|
|
|
|
|
| 9707028061 |
a_x = 0
|
|
|
|
|
| 2741489181 |
a_y = -g
|
|
|
|
|
| 3270039798 |
2
|
|
|
|
|
| 8880467139 |
2
|
|
|
|
|
| 8750379055 |
0 = \frac{d}{dt} v_x
|
|
|
|
|
| 1977955751 |
-g = \frac{d}{dt} v_y
|
|
|
|
|
| 6672141531 |
dt
|
|
|
|
|
| 1702349646 |
-g dt = d v_y
|
|
|
|
|
| 8584698994 |
-g \int dt = \int d v_y
|
|
|
|
|
| 9973952056 |
-g t = v_y - v_{0, y}
|
|
|
|
|
| 4167526462 |
v_{0, y}
|
|
|
|
|
| 6572039835 |
-g t + v_{0, y} = v_y
|
|
|
|
|
| 7252338326 |
v_y = \frac{dy}{dt}
|
|
|
|
|
| 6204539227 |
-g t + v_{0, y} = \frac{dy}{dt}
|
|
|
|
|
| 1614343171 |
dt
|
|
|
|
|
| 8145337879 |
-g t dt + v_{0, y} dt = dy
|
|
|
|
|
| 8808860551 |
-g \int t dt + v_{0, y} \int dt = \int dy
|
|
|
|
|
| 2858549874 |
- \frac{1}{2} g t^2 + v_{0, y} t = y - y_0
|
|
|
|
|
| 6098638221 |
y_0
|
|
|
|
|
| 2461349007 |
- \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y
|
|
|
|
|
| 8717193282 |
dt
|
|
|
|
|
| 1166310428 |
0 dt = d v_x
|
|
|
|
|
| 2366691988 |
\int 0 dt = \int d v_x
|
|
|
|
|
| 1676472948 |
0 = v_x - v_{0, x}
|
|
|
|
|
| 1439089569 |
v_{0, x}
|
|
|
|
|
| 6134836751 |
v_{0, x} = v_x
|
|
|
|
|
| 8460820419 |
v_x = \frac{dx}{dt}
|
|
|
|
|
| 7455581657 |
v_{0, x} = \frac{dx}{dt}
|
|
|
|
|
| 8607458157 |
dt
|
|
|
|
|
| 1963253044 |
v_{0, x} dt = dx
|
|
|
|
|
| 3676159007 |
v_{0, x} \int dt = \int dx
|
|
|
|
|
| 9882526611 |
v_{0, x} t = x - x_0
|
|
|
|
|
| 3182907803 |
x_0
|
|
|
|
|
| 8486706976 |
v_{0, x} t + x_0 = x
|
|
|
|
|
| 1306360899 |
x = v_{0, x} t + x_0
|
|
|
|
|
| 7049769409 |
2
|
|
|
|
|
| 9341391925 |
\vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y}
|
|
|
|
|
| 6410818363 |
\theta
|
|
|
|
|
| 7391837535 |
\cos(\theta) = \frac{v_{0, x}}{v_0}
|
|
|
|
|
| 7376526845 |
\sin(\theta) = \frac{v_{0, y}}{v_0}
|
|
|
|
|
| 5868731041 |
v_0
|
|
|
|
|
| 6083821265 |
v_0 \cos(\theta) = v_{0, x}
|
|
|
|
|
| 5438722682 |
x = v_0 t \cos(\theta) + x_0
|
|
|
|
|
| 5620558729 |
v_0
|
|
|
|
|
| 8949329361 |
v_0 \sin(\theta) = v_{0, y}
|
|
|
|
|
| 1405465835 |
y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0
|
|
|
|
|
| 9862900242 |
y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0
|
|
|
|
|
| 6050070428 |
v_{0, x}
|
|
|
|
|
| 3274926090 |
t = \frac{x - x_0}{v_{0, x}}
|
|
|
|
|
| 7354529102 |
y = - \frac{1}{2} g \left( \frac{x - x_0}{v_{0, x}} \right)^2 + v_{0, y} \frac{x - x_0}{v_{0, x}} + y_0
|
|
|
|
|
| 8406170337 |
y \to y_f
|
|
|
|
|
| 2403773761 |
t \to t_f
|
|
|
|
|
| 5379546684 |
y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
|
|
|
|
|
| 5373931751 |
t = t_f
|
|
|
|
|
| 9112191201 |
y_f = 0
|
|
|
|
|
| 8198310977 |
0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
|
|
|
|
|
| 1650441634 |
y_0 = 0
|
|
|
define coordinate system such that initial height is at origin
|
|
| 1087417579 |
0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)
|
|
|
|
|
| 4829590294 |
t_f
|
|
|
|
|
| 2086924031 |
0 = - \frac{1}{2} g t_f + v_0 \sin(\theta)
|
|
|
|
|
| 6974054946 |
\frac{1}{2} g t_f
|
|
|
|
|
| 1191796961 |
\frac{1}{2} g t_f = v_0 \sin(\theta)
|
|
|
|
|
| 2510804451 |
2/g
|
|
|
|
|
| 4778077984 |
t_f = \frac{2 v_0 \sin(\theta)}{g}
|
|
|
|
|
| 3273630811 |
x \to x_f
|
|
|
|
|
| 6732786762 |
t \to t_f
|
|
|
|
|
| 3485125659 |
x_f = v_0 t_f \cos(\theta) + x_0
|
|
|
|
|
| 4370074654 |
t = t_f
|
|
|
|
|
| 2378095808 |
x_f = x_0 + d
|
|
|
|
|
| 4268085801 |
x_0 + d = v_0 t_f \cos(\theta) + x_0
|
|
|
|
|
| 8072682558 |
x_0
|
|
|
|
|
| 7233558441 |
d = v_0 t_f \cos(\theta)
|
|
|
|
|
| 2297105551 |
d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)
|
|
|
|
|
| 7587034465 |
\theta
|
|
|
|
|
| 7214442790 |
x
|
|
|
|
|
| 2519058903 |
\sin(2 \theta) = 2 \sin(\theta) \cos(\theta)
|
|
|
|
|
| 8922441655 |
d = \frac{v_0^2}{g} \sin(2 \theta)
|
|
|
|
|
| 5667870149 |
\theta
|
|
|
|
|
| 1541916015 |
\theta = \frac{\pi}{4}
|
|
|
|
|
| 3607070319 |
d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)
|
|
|
|
|
| 5353282496 |
d = \frac{v_0^2}{g}
|
|
|
|
|
| 0000040490 |
a^2
|
|
|
|
|
| 0000999900 |
b/(2a)
|
|
|
|
|
| 0001030901 |
\cos(x)
|
|
|
|
|
| 0001111111 |
(\sin(x))^2
|
|
|
|
|
| 0001209482 |
2 \pi
|
|
|
|
|
| 0001304952 |
\hbar
|
|
|
|
|
| 0001334112 |
W
|
|
|
|
|
| 0001921933 |
2 i
|
|
|
|
|
| 0002239424 |
2
|
|
|
|
|
| 0002338514 |
\vec{p}_{2}
|
|
|
|
|
| 0002342425 |
m/m
|
|
|
|
|
| 0002367209 |
1/2
|
|
|
|
|
| 0002393922 |
x
|
|
|
|
|
| 0002424922 |
a
|
|
|
|
|
| 0002436656 |
i \hbar
|
|
|
|
|
| 0002449291 |
b/(2a)
|
|
|
|
|
| 0002838490 |
b/(2a)
|
|
|
|
|
| 0002919191 |
\sin(-x)
|
|
|
|
|
| 0002929944 |
1/2
|
|
|
|
|
| 0002940021 |
2 \pi
|
|
|
|
|
| 0003232242 |
t
|
|
|
|
|
| 0003413423 |
\cos(-x)
|
|
|
|
|
| 0003747849 |
-1
|
|
|
|
|
| 0003838111 |
2
|
|
|
|
|
| 0003919391 |
x
|
|
|
|
|
| 0003949052 |
-x
|
|
|
|
|
| 0003949921 |
\hbar
|
|
|
|
|
| 0003954314 |
dx
|
|
|
|
|
| 0003981813 |
-\sin(x)
|
|
|
|
|
| 0004089571 |
2x
|
|
|
|
|
| 0004264724 |
y
|
|
|
|
|
| 0004307451 |
(b/(2a))^2
|
|
|
|
|
| 0004582412 |
x
|
|
|
|
|
| 0004829194 |
2
|
|
|
|
|
| 0004831494 |
a
|
|
|
|
|
| 0004849392 |
x
|
|
|
|
|
| 0004858592 |
h
|
|
|
|
|
| 0004934845 |
x
|
|
|
|
|
| 0004948585 |
a
|
|
|
|
|
| 0005395034 |
a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
|
|
|
|
|
| 0005626421 |
t
|
|
|
|
|
| 0005749291 |
f
|
|
|
|
|
| 0005938585 |
\frac{-\hbar^2}{2m}
|
|
|
|
|
| 0006466214 |
(\sin(x))^2
|
|
|
|
|
| 0006544644 |
t
|
|
|
|
|
| 0006563727 |
x
|
|
|
|
|
| 0006644853 |
c/a
|
|
|
|
|
| 0006656532 |
e
|
|
|
|
|
| 0007471778 |
2(\sin(x))^2
|
|
|
|
|
| 0007563791 |
i
|
|
|
|
|
| 0007636749 |
x
|
|
|
|
|
| 0007894942 |
(\sin(x))^2
|
|
|
|
|
| 0008837284 |
T
|
|
|
|
|
| 0008842811 |
\cos(2x)
|
|
|
|
|
| 0009458842 |
\psi(x)
|
|
|
|
|
| 0009484724 |
\frac{n \pi}{W}x
|
|
|
|
|
| 0009485857 |
a^2\frac{2}{W}
|
|
|
|
|
| 0009485858 |
\frac{2n\pi}{W}
|
|
|
|
|
| 0009492929 |
v du
|
|
|
|
|
| 0009587738 |
\psi
|
|
|
|
|
| 0009594995 |
1/2
|
|
|
|
|
| 0009877781 |
y
|
|
|
|
|
| 0203024440 |
1 = \int_0^W a \sin(\frac{n \pi}{W} x) \psi(x)^* dx
|
|
|
|
|
| 0404050504 |
\lambda = \frac{v}{f}
|
|
|
|
|
| 0439492440 |
\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)\right|_0^W
|
|
|
|
|
| 0934990943 |
k = \frac{2 \pi}{v T}
|
|
|
|
|
| 0948572140 |
\int \cos(a x) dx = \frac{1}{a}\sin(a x)
|
|
|
|
|
| 1010393913 |
\langle \psi| \hat{A}^+ |\psi \rangle = \langle a \rangle^*
|
|
|
|
|
| 1010393944 |
x = \langle\psi_{\alpha}| a_{\beta} |\psi_{\beta} \rangle
|
|
|
|
|
| 1010923823 |
k W = n \pi
|
|
|
|
|
| 1020010291 |
0 = a \sin(k W)
|
|
|
|
|
| 1020394900 |
p = h/\lambda
|
|
|
|
|
| 1020394902 |
E = h f
|
|
|
|
|
| 1029039903 |
p = m v
|
|
|
|
|
| 1029039904 |
p^2 = m^2 v^2
|
|
|
|
|
| 1158485859 |
\frac{-\hbar^2}{2m} \nabla^2 = {\cal H}
|
|
|
|
|
| 1202310110 |
\frac{1}{a^2} = \int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
|
|
|
|
|
| 1202312210 |
\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
|
|
|
|
|
| 1203938249 |
a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle = a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
|
|
|
|
|
| 1248277773 |
\cos(2x) = 1-2(\sin(x))^2
|
|
|
|
|
| 1293913110 |
0 = b
|
|
|
|
|
| 1293923844 |
\lambda = v T
|
|
|
|
|
| 1314464131 |
\vec{\nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
|
|
|
|
|
| 1314864131 |
\vec{\nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E}
|
|
|
|
|
| 1395858355 |
x = \langle \psi_{\alpha}| a_{\alpha} |\psi_{\beta}\rangle
|
|
|
|
|
| 1492811142 |
f = x - d
|
|
|
|
|
| 1492842000 |
\nabla \vec{x} = f(y)
|
|
|
|
|
| 1636453295 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = - \nabla^2 \vec{E}
|
|
|
|
|
| 1638282134 |
\vec{p}_{before} = \vec{p}_{after}
|
|
|
|
|
| 1648958381 |
\nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{\nabla}\psi(\vec{r},t)\right)
|
|
|
|
|
| 1857710291 |
0 = a \sin(n \pi)
|
|
|
|
|
| 1858578388 |
\nabla^2 E(\vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 1858772113 |
k = \frac{n \pi}{W}
|
|
|
|
|
| 1934748140 |
\int |\psi(x)|^2 dx = 1
|
|
|
|
|
| 2029293929 |
\nabla^2 E(\vec{r})\exp(i \omega t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 2103023049 |
\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right)
|
|
|
|
|
| 2113211456 |
f = 1/T
|
|
|
|
|
| 2123139121 |
-\exp(-i x) = -\cos(x)+i \sin(x)
|
|
|
|
|
| 2131616531 |
T f = 1
|
|
|
|
|
| 2258485859 |
{\cal H} \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
|
|
|
|
|
| 2394240499 |
x = a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle
|
|
|
|
|
| 2394853829 |
\exp(-i x) = \cos(-x)+i \sin(-x)
|
|
|
|
|
| 2394935831 |
( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
|
|
|
|
|
| 2394935835 |
\left(\langle\psi| \hat{A} |\psi\rangle \right)^+ = \left(\langle a \rangle\right)^+
|
|
|
|
|
| 2395958385 |
\nabla^2 \psi\left(\vec{r},t)\right) = \frac{-p^2}{\hbar} \psi(\vec{r},t)
|
|
|
|
|
| 2404934990 |
\langle x^2\rangle -2\langle x \rangle\langle x \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 2494533900 |
\langle x^2\rangle -\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 2569154141 |
\vec{\nabla} \times \frac{\partial}{\partial t}\vec{H} = \epsilon_0 \frac{\partial^2 }{\partial t^2}\vec{E}
|
|
|
|
|
| 2648958382 |
\nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p}\psi(\vec{r},t)\right)
|
|
|
|
|
| 2848934890 |
\langle a \rangle^* = \langle a \rangle
|
|
|
|
|
| 2944838499 |
\psi(x) = a \sin(\frac{n \pi}{W} x)
|
|
|
|
|
| 3121234211 |
\frac{k}{2\pi} = \lambda
|
|
|
|
|
| 3121234212 |
p = \frac{h k}{2\pi}
|
|
|
|
|
| 3121513111 |
k = \frac{2 \pi}{\lambda}
|
|
|
|
|
| 3131111133 |
T = 1 / f
|
|
|
|
|
| 3131211131 |
\omega = 2 \pi f
|
|
|
|
|
| 3132131132 |
\omega = \frac{2\pi}{T}
|
|
|
|
|
| 3147472131 |
\frac{\omega}{2 \pi} = f
|
|
|
|
|
| 3285732911 |
(\cos(x))^2 = 1-(\sin(x))^2
|
|
|
|
|
| 3485475729 |
\nabla^2 E(\vec{r}) = - \frac{\omega^2}{c^2} E(\vec{r})
|
|
|
|
|
| 3585845894 |
\langle \left(x-\langle x \rangle\right)^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 3829492824 |
\frac{1}{2}\left(\exp(i x)+\exp(-i x)\right) = \cos(x)
|
|
|
|
|
| 3924948349 |
a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
|
|
|
|
|
| 3942849294 |
\exp(i x)-\exp(-i x) = 2 i \sin(x)
|
|
|
|
|
| 3943939590 |
x = a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle
|
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|
| 3947269979 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
|
|
|
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|
| 3948571256 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi(\vec{r},t)
|
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|
| 3948572230 |
\vec{\nabla}\psi(\vec{r},t) = \psi_0 \vec{\nabla}\exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
|
|
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|
| 3948574224 |
\psi(\vec{r},t) = \psi_0 \exp\left(i\left(\vec{k}\cdot\vec{r} - \omega t \right) \right)
|
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|
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|
| 3948574226 |
\psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right)
|
|
|
|
|
| 3948574228 |
\psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
|
|
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|
|
| 3948574230 |
\psi(\vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
|
|
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|
|
| 3948574233 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
|
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|
| 3948574235 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
|
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| 3951205425 |
\vec{p}_{after} = \vec{p}_{1}
|
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|
| 4147472132 |
E = \frac{h \omega}{2 \pi}
|
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| 4298359835 |
E = \frac{1}{2}m v^2
|
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|
|
| 4298359845 |
E = \frac{1}{2m}m^2 v^2
|
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|
|
| 4298359851 |
E = \frac{p^2}{2m}
|
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|
|
|
| 4341171256 |
i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{p^2}{2 m} \psi(\vec{r},t)
|
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|
|
|
| 4348571256 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi(\vec{r},t)
|
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|
| 4394958389 |
\vec{\nabla}\cdot \left(\vec{\nabla}\psi(\vec{r},t)\right) = \frac{i}{\hbar} \vec{\nabla}\cdot\left( \vec{p} \psi(\vec{r},t)\right)
|
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|
| 4585828572 |
\epsilon_0 \mu_0 = \frac{1}{c^2}
|
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|
| 4585932229 |
\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
|
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|
| 4598294821 |
\exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
|
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|
|
| 4638429483 |
\exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))
|
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|
|
| 4742644828 |
\exp(i x)+\exp(-i x) = 2 \cos(x)
|
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|
| 4827492911 |
\cos(2x)+(\sin(x))^2 = 1-(\sin(x))^2
|
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|
| 4838429483 |
\exp(2 i x) = \cos(2 x)+i \sin(2 x)
|
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|
| 4843995999 |
\frac{1}{2 i}\left(\exp(i x)-\exp(-i x)\right) = \sin(x)
|
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|
| 4857472413 |
1 = \int \psi(x)\psi(x)^* dx
|
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|
| 4857475848 |
\frac{1}{a^2} = \frac{W}{2}
|
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|
|
| 4858429483 |
\exp(i x)\exp(i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))
|
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|
|
|
| 4923339482 |
i x = \log(y)
|
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|
| 4928239482 |
\log(y) = i x
|
|
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|
|
| 4928923942 |
a = b
|
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|
| 4929218492 |
a+b = c
|
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|
| 4929482992 |
b = 2
|
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|
|
| 4938429482 |
\exp(-i x) = \cos(x)+i \sin(-x)
|
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|
|
|
| 4938429483 |
\exp(i x) = \cos(x)+i \sin(x)
|
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|
|
| 4938429484 |
\exp(-i x) = \cos(x)-i \sin(x)
|
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|
| 4943571230 |
\vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
|
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|
|
|
| 4948934890 |
\langle \psi| \hat{A} |\psi\rangle = \langle a \rangle^*
|
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|
|
| 4949359835 |
\langle x^2\rangle -2\langle x^2 \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 4954839242 |
\cos(2x)+i\sin(2x) = (\cos(x)+i \sin(x))(\cos(x)+i \sin(x))
|
|
|
|
|
| 4978429483 |
\exp(-i x) = \cos(-x)+i \sin(-x)
|
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|
|
| 4984892984 |
a+2 = c
|
|
|
|
|
| 4985825552 |
\nabla^2 E(\vec{r})\exp(i \omega t) = i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 5530148480 |
\vec{p}_{1}-\vec{p}_{2} = \vec{p}_{electron}
|
|
|
|
|
| 5727578862 |
\frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)
|
|
|
|
|
| 5832984291 |
(\sin(x))^2 + (\cos(x))^2 = 1
|
|
|
|
|
| 5857434758 |
\int a dx = a x
|
|
|
|
|
| 5868688585 |
\frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = \frac{p^2}{2m} \psi(\vec{r},t)
|
|
|
|
|
| 5900595848 |
k = \frac{\omega}{v}
|
|
|
|
|
| 5928285821 |
x^2 + 2x(b/(2a)) + (b/(2a))^2 = (x+(b/(2a)))^2
|
|
|
|
|
| 5928292841 |
x^2 + (b/a)x + (b/(2a))^2 = -c/a + (b/(2a))^2
|
|
|
|
|
| 5938459282 |
x^2 + (b/a)x = -c/a
|
|
|
|
|
| 5958392859 |
x^2 + (b/a)x+(c/a) = 0
|
|
|
|
|
| 5959282914 |
x^2 + x(b/a) + (b/(2a))^2 = (x+(b/(2a)))^2
|
|
|
|
|
| 5982958248 |
x = -\sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
|
|
|
|
|
| 5982958249 |
x+(b/(2a)) = -\sqrt{(b/(2a))^2 - (c/a)}
|
|
|
|
|
| 5985371230 |
\vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi(\vec{r},t)
|
|
|
|
|
| 6742123016 |
\vec{p}_{electron}\cdot\vec{p}_{electron} = (\vec{p}_{1}\cdot\vec{p}_{1})+(\vec{p}_{2}\cdot\vec{p}_{2})-2(\vec{p}_{1}\cdot\vec{p}_{2})
|
|
|
|
|
| 7466829492 |
\vec{\nabla} \cdot \vec{E} = 0
|
|
|
|
|
| 7564894985 |
\int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)
|
|
|
|
|
| 7572664728 |
\cos(2x)+2(\sin(x))^2 = 1
|
|
|
|
|
| 7575738420 |
\left(\sin\left(\frac{n \pi}{W}x\right)\right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2}
|
|
|
|
|
| 7575859295 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859300 |
\epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{\nabla} \times \vec{E} )_k = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859302 |
\epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859304 |
\epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h}
|
|
|
|
|
| 7575859306 |
\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859308 |
\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right) = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859310 |
\hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859312 |
\vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7917051060 |
\vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2}
|
|
|
|
|
| 8139187332 |
\vec{p}_{1} = \vec{p}_{2}+\vec{p}_{electron}
|
|
|
|
|
| 8257621077 |
\vec{p}_{before} = \vec{p}_{1}
|
|
|
|
|
| 8311458118 |
\vec{p}_{after} = \vec{p}_{2}+\vec{p}_{electron}
|
|
|
|
|
| 8399484849 |
\langle x^2 -2x\langle x \rangle+\langle x \rangle^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 8484544728 |
-a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x)
|
|
|
|
|
| 8485757728 |
a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(kx)
|
|
|
|
|
| 8485867742 |
\frac{2}{W} = a^2
|
|
|
|
|
| 8489593958 |
d(u v) = u dv + v du
|
|
|
|
|
| 8489593960 |
d(u v) - v du = u dv
|
|
|
|
|
| 8489593962 |
u dv = d(u v) - v du
|
|
|
|
|
| 8489593964 |
\int u dv = u v - \int v du
|
|
|
|
|
| 8494839423 |
\nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
|
|
|
|
|
| 8572657110 |
1 = \int |\psi(x)|^2 dx
|
|
|
|
|
| 8572852424 |
\vec{E} = E(\vec{r},t)
|
|
|
|
|
| 8575746378 |
\int \frac{1}{2} dx = \frac{1}{2} x
|
|
|
|
|
| 8575748999 |
\frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x)\right) = -k^2 \left(a \sin(kx) + b \cos(kx)\right)
|
|
|
|
|
| 8576785890 |
1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
|
|
|
|
|
| 8577275751 |
0 = a \sin(0) + b\cos(0)
|
|
|
|
|
| 8582885111 |
\psi(x) = a \sin(kx) + b \cos(kx)
|
|
|
|
|
| 8582954722 |
x^2 + 2xh + h^2 = (x+h)^2
|
|
|
|
|
| 8849289982 |
\psi(x)^* = a \sin(\frac{n \pi}{W} x)
|
|
|
|
|
| 8889444440 |
1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right)\right)^2 dx
|
|
|
|
|
| 9059289981 |
\psi(x) = a \sin(k x)
|
|
|
|
|
| 9285928292 |
ax^2 + bx + c = 0
|
|
|
|
|
| 9291999979 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0\vec{\nabla} \times \frac{\partial \vec{H}}{\partial t}
|
|
|
|
|
| 9294858532 |
\hat{A}^+ = \hat{A}
|
|
|
|
|
| 9385938295 |
(x+(b/(2a)))^2 = -(c/a) + (b/(2a))^2
|
|
|
|
|
| 9393939991 |
\psi(x) = -\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
|
|
|
|
|
| 9393939992 |
\psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
|
|
|
|
|
| 9394939493 |
\nabla^2 E(\vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r},t)
|
|
|
|
|
| 9429829482 |
\frac{d}{dx} y = -\sin(x) + i\cos(x)
|
|
|
|
|
| 9482113948 |
\frac{dy}{y} = i dx
|
|
|
|
|
| 9482438243 |
(\cos(x))^2 = \cos(2x)+(\sin(x))^2
|
|
|
|
|
| 9482923849 |
\exp(i x) = y
|
|
|
|
|
| 9482928242 |
\cos(2x) = (\cos(x))^2 - (\sin(x))^2
|
|
|
|
|
| 9482928243 |
\cos(2x)+(\sin(x))^2 = (\cos(x))^2
|
|
|
|
|
| 9482943948 |
\log(y) = i dx
|
|
|
|
|
| 9482948292 |
b = c
|
|
|
|
|
| 9482984922 |
\frac{d}{dx} y = (i\sin(x) + \cos(x)) i
|
|
|
|
|
| 9483928192 |
\cos(2x)+i\sin(2x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
|
|
|
|
|
| 9485384858 |
\nabla^2 E(\vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 9485747245 |
\sqrt{\frac{2}{W}} = a
|
|
|
|
|
| 9485747246 |
-\sqrt{\frac{2}{W}} = a
|
|
|
|
|
| 9492920340 |
y = \cos(x)+i \sin(x)
|
|
|
|
|
| 9494829190 |
k
|
|
|
|
|
| 9495857278 |
\psi(x=W) = 0
|
|
|
|
|
| 9499428242 |
E(\vec{r},t) = E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 9499959299 |
a + k = b + k
|
|
|
|
|
| 9582958293 |
x = \sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
|
|
|
|
|
| 9582958294 |
x+(b/(2a)) = \sqrt{(b/(2a))^2 - (c/a)}
|
|
|
|
|
| 9584821911 |
c + d = a
|
|
|
|
|
| 9585727710 |
\psi(x=0) = 0
|
|
|
|
|
| 9596004948 |
x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle
|
|
|
|
|
| 9848292229 |
dy = y i dx
|
|
|
|
|
| 9848294829 |
\frac{d}{dx} y = y i
|
|
|
|
|
| 9858028950 |
\frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
|
|
|
|
|
| 9889984281 |
2(\sin(x))^2 = 1-\cos(2x)
|
|
|
|
|
| 9919999981 |
\rho = 0
|
|
|
|
|
| 9958485859 |
\frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
|
|
|
|
|
| 9988949211 |
(\sin(x))^2 = \frac{1-\cos(2x)}{2}
|
|
|
|
|
| 9991999979 |
\vec{\nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t}
|
|
|
|
|
| 9999998870 |
\frac{\vec{p}}{\hbar} = \vec{k}
|
|
|
|
|
| 9999999870 |
\frac{p}{\hbar} = k
|
|
|
|
|
| 9999999950 |
\beta = 1/(k_{Boltzmann}T)
|
|
|
|
|
| 9999999951 |
\langle x | k \rangle = \frac{\exp(i k x)}{\sqrt{2\pi}}
|
|
|
|
|
| 9999999952 |
f(x) \delta(x-a) = f(a)
|
|
|
|
|
| 9999999953 |
\int_{-\infty}^{\infty} \delta(x) dx = 1
|
|
|
|
|
| 9999999954 |
c = 1/\sqrt{\epsilon_0 \mu_0}
|
|
|
|
|
| 9999999955 |
\vec{E} = -\vec{\nabla} \Psi
|
|
|
|
|
| 9999999956 |
\vec{F} = \frac{\partial}{\partial t}\vec{p}
|
|
|
|
|
| 9999999957 |
\vec{F} = -\vec{\nabla} V
|
|
|
|
|
| 9999999960 |
\hbar = h/(2 \pi)
|
|
|
|
|
| 9999999961 |
\frac{E}{\hbar} = \omega
|
|
|
|
|
| 9999999962 |
p = \hbar k
|
|
|
|
|
| 9999999963 |
\lambda = h/p
|
|
|
|
|
| 9999999964 |
\omega = c k
|
|
|
|
|
| 9999999965 |
E = \omega \hbar
|
|
|
|
|
| 9999999966 |
\vec{L} = \vec{r}\times\vec{p}
|
|
|
|
|
| 9999999967 |
\vec{S} = \frac{1}{\mu_0} \vec{E}\times \vec{B}
|
|
|
|
|
| 9999999968 |
x = \frac{-b-\sqrt{b^2-4ac}}{2a}
|
|
|
|
|
| 9999999969 |
x = \frac{-b+\sqrt{b^2-4ac}}{2a}
|
|
|
|
|
| 9999999970 |
\eta_1 \sin\theta_1 = \eta_2 \sin\theta_2
|
|
|
|
|
| 9999999971 |
{\cal H} = \frac{p^2}{2m}+V
|
|
|
|
|
| 9999999972 |
{\cal H} |\psi\rangle = E |\psi\rangle
|
|
|
|
|
| 9999999973 |
\left( \Delta A \right)^2 = \langle A^2 \rangle - \langle A \rangle^2
|
|
|
|
|
| 9999999974 |
\langle \psi| \hat{A} |\psi\rangle = \int_{-\infty}^{\infty} \psi^* A \psi dx
|
|
|
|
|
| 9999999975 |
\langle \psi| \hat{A} |\psi\rangle = \langle a \rangle
|
|
|
|
|
| 9999999976 |
\hat{p} = -i \hbar \frac{\partial }{\partial x}
|
|
|
|
|
| 9999999977 |
[\hat{x},\hat{p}] = i \hbar
|
|
|
|
|
| 9999999978 |
\vec{\nabla} \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial E}{\partial t}
|
|
|
|
|
| 9999999979 |
\vec{\nabla} \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}
|
|
|
|
|
| 9999999980 |
\vec{\nabla} \cdot \vec{B} = 0
|
|
|
|
|
| 9999999981 |
\vec{\nabla} \cdot \vec{E} = \rho/\epsilon_0
|
|
|
|
|
| 9999999982 |
V = I R + Q/C + L \frac{\partial I}{\partial t}
|
|
|
|
|
| 9999999983 |
C = V A/d
|
|
|
|
|
| 9999999984 |
Q = C V
|
|
|
|
|
| 9999999985 |
V = I R
|
|
|
|
|
| 9999999986 |
\left[\right]\left[\right] = \left[\right]
|
|
|
|
|
| 9999999987 |
1 atmosphere = 101.325 Pascal
|
|
|
|
|
| 9999999988 |
1 atmosphere = 14.7 pounds/(inch^2)
|
|
|
|
|
| 9999999989 |
mass_{electron} = 511000 electronVolts/(q^2)
|
|
|
|
|
| 9999999990 |
Tesla = 10000*Gauss
|
|
|
|
|
| 9999999991 |
Pascal = Newton / (meter^2)
|
|
|
|
|
| 9999999992 |
Tesla = Newton / (Ampere*meter)
|
|
|
|
|
| 9999999993 |
Farad = Coulumb / Volt
|
|
|
|
|
| 9999999995 |
Volt = Joule / Coulumb
|
|
|
|
|
| 9999999996 |
Coulomb = Ampere / second
|
|
|
|
|
| 9999999997 |
Watt = Joule / second
|
|
|
|
|
| 9999999998 |
Joule = Newton*meter
|
|
|
|
|
| 9999999999 |
Newton = kilogram*meter/(second^2)
|
|
|
|
|
| 4961662865 |
x
|
|
|
|
|
| 9110536742 |
2 x
|
|
|
|
|
| 8483686863 |
\sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x)\right)
|
|
|
|
|
| 3470587782 |
\sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
|
|
|
|
|
| 8642992037 |
2
|
|
|
|
|
| 9894826550 |
2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \left(\exp(i x)+\exp(-i x)\right)
|
|
|
|
|
| 4257214632 |
\frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)
|
|
|
|
|
| 8699789241 |
2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)
|
|
|
|
|
| 9180861128 |
2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)
|
|
|
|
|
| 2405307372 |
\sin(2 x) = 2 \sin(x) \cos(x)
|
|
|
|
|
| 3268645065 |
x
|
|
|
|
|
| 9350663581 |
\pi
|
|
|
|
|
| 8332931442 |
\exp(i \pi) = \cos(\pi)+i \sin(\pi)
|
|
|
|
|
| 6885625907 |
\exp(i \pi) = -1 + i 0
|
|
|
|
|
| 3331824625 |
\exp(i \pi) = -1
|
|
|
|
|
| 4901237716 |
1
|
|
|
|
|
| 2501591100 |
\exp(i \pi) + 1 = 0
|
|
|
|
|
| 5136652623 |
E = KE + PE
|
|
mechanical energy is the sum of the potential plus kinetic energies |
|
|
| 7915321076 |
E, KE, PE
|
|
|
|
|
| 3192374582 |
E_2, KE_2, PE_2
|
|
|
|
|
| 7875206161 |
E_2 = KE_2 + PE_2
|
|
|
|
|
| 4229092186 |
E, KE, PE
|
|
|
|
|
| 1490821948 |
E_1, KE_1, PE_1
|
|
|
|
|
| 4303372136 |
E_1 = KE_1 + PE_1
|
|
|
|
|
| 5514556106 |
E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1)
|
|
|
|
|
| 8357234146 |
KE = \frac{1}{2} m v^2
|
|
kinetic energy |
Kinetic_energy
|
|
| 3658066792 |
KE_2, v_2
|
|
|
|
|
| 8082557839 |
KE, v
|
|
|
|
|
| 7676652285 |
KE_2 = \frac{1}{2} m v_2^2
|
|
|
|
|
| 2790304597 |
KE_1, v_1
|
|
|
|
|
| 9880327189 |
KE, v
|
|
|
|
|
| 4928007622 |
KE_1 = \frac{1}{2} m v_1^2
|
|
|
|
|
| 5733146966 |
KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right)
|
|
|
|
|
| 6554292307 |
t
|
|
|
|
|
| 4270680309 |
\frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t}
|
|
|
|
|
| 5781981178 |
x^2 - y^2 = (x+y)(x-y)
|
|
difference of squares |
Difference_of_two_squares
|
|
| 5946759357 |
v_2, v_1
|
|
|
|
|
| 6675701064 |
x, y
|
|
|
|
|
| 4648451961 |
v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1)
|
|
|
|
|
| 9356924046 |
\frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t}
|
|
|
|
|
| 9397152918 |
v = \frac{v_1 + v_2}{2}
|
|
average velocity |
|
|
| 2857430695 |
a = \frac{v_2 - v_1}{t}
|
|
acceleration |
|
|
| 7735737409 |
\frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t}
|
|
|
|
|
| 4784793837 |
\frac{KE_2 - KE_1}{t} = m v a
|
|
|
|
|
| 5345738321 |
F = m a
|
|
Newton's second law of motion |
Newton%27s_laws_of_motion#Newton's_second_law
|
|
| 2186083170 |
\frac{KE_2 - KE_1}{t} = v F
|
|
|
|
|
| 6715248283 |
PE = -F x
|
|
potential energy |
Potential_energy
|
|
| 3852078149 |
PE_2, x_2
|
|
|
|
|
| 7388921188 |
PE, x
|
|
|
|
|
| 2431507955 |
PE_2 = -F x_2
|
|
|
|
|
| 3412641898 |
PE_1, x_1
|
|
|
|
|
| 9937169749 |
PE, x
|
|
|
|
|
| 4669290568 |
PE_1 = -F x_1
|
|
|
|
|
| 7734996511 |
PE_2 - PE_1 = -F ( x_2 - x_1 )
|
|
|
|
|
| 2016063530 |
t
|
|
|
|
|
| 7267155233 |
\frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right)
|
|
|
|
|
| 9337785146 |
v = \frac{x_2 - x_1}{t}
|
|
average velocity |
|
|
| 4872970974 |
\frac{PE_2 - PE_1}{t} = -F v
|
|
|
|
|
| 2081689540 |
t
|
|
|
|
|
| 2770069250 |
\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t}
|
|
|
|
|
| 3591237106 |
\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v
|
|
|
|
|
| 1772416655 |
\frac{E_2 - E_1}{t} = v F - F v
|
|
|
|
|
| 1809909100 |
\frac{E_2 - E_1}{t} = 0
|
|
|
|
|
| 5778176146 |
t
|
|
|
|
|
| 3806977900 |
E_2 - E_1 = 0
|
|
|
|
|
| 5960438249 |
E_1
|
|
|
|
|
| 8558338742 |
E_2 = E_1
|
|
|
|
|
| 8945218208 |
\theta_{Brewster} + \theta_{refracted} = 90^{\circ}
|
|
|
based on figure 34-27 on page 824 in 2001_HRW
|
|
| 9025853427 |
\theta_{Brewster}
|
|
|
|
|
| 1310571337 |
\theta_{refracted} = 90^{\circ} - \theta_{Brewster}
|
|
|
|
|
| 6450985774 |
n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 )
|
|
Law of Refraction |
eq 34-44 on page 819 in 2001_HRW
|
|
| 1779497048 |
\theta_{Brewster}, \theta_{refraction}
|
|
|
|
|
| 7322344455 |
\theta_1, \theta_2
|
|
|
|
|
| 2575937347 |
n_1 \sin( \theta_{Brewster} ) = n_2 \sin( \theta_{refraction} )
|
|
|
|
|
| 7696214507 |
n_1 \sin( \theta_{Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{Brewster} )
|
|
|
|
|
| 8588429722 |
\sin( 90^{\circ} - x ) = \cos( x )
|
|
|
|
|
| 7375348852 |
\theta_{Brewster}
|
|
|
|
|
| 1512581563 |
x
|
|
|
|
|
| 6831637424 |
\sin( 90^{\circ} - \theta_{Brewster} ) = \cos( \theta_{Brewster} )
|
|
|
|
|
| 3061811650 |
n_1 \sin( \theta_{Brewster} ) = n_2 \cos( \theta_{Brewster} )
|
|
|
|
|
| 7857757625 |
n_1
|
|
|
|
|
| 9756089533 |
\sin( \theta_{Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{Brewster} )
|
|
|
|
|
| 5632428182 |
\cos( \theta_{Brewster} )
|
|
|
|
|
| 2768857871 |
\frac{\sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )} = \frac{n_2}{n_1}
|
|
|
|
|
| 4968680693 |
\tan( x ) = \frac{ \sin( x )}{\cos( x )}
|
|
|
|
|
| 7321695558 |
\theta_{Brewster}
|
|
|
|
|
| 9906920183 |
x
|
|
|
|
|
| 4501377629 |
\tan( \theta_{Brewster} ) = \frac{ \sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )}
|
|
|
|
|
| 3417126140 |
\tan( \theta_{Brewster} ) = \frac{ n_2 }{ n_1 }
|
|
|
|
|
| 5453995431 |
\arctan{ x }
|
|
|
|
|
| 6023986360 |
x
|
|
|
|
|
| 8495187962 |
\theta_{Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }
|
|
|
|
|
| 9040079362 |
f
|
|
|
|
|
| 9565166889 |
T
|
|
|
|
|
| 5426308937 |
v = \frac{d}{t}
|
|
|
|
|
| 7476820482 |
C
|
|
|
|
|
| 1277713901 |
d
|
|
|
|
|
| 6946088325 |
v = \frac{C}{t}
|
|
|
|
|
| 6785303857 |
C = 2 \pi r
|
|
|
|
|
| 4816195267 |
C, r
|
|
|
|
|
| 2113447367 |
C_{\rm{Earth\ orbit}}, r_{\rm{Earth\ orbit}}
|
|
|
|
|
| 6348260313 |
C_{\rm{Earth\ orbit}} = 2 \pi r_{\rm{Earth\ orbit}}
|
|
|
|
|
| 3847777611 |
C, v, t
|
|
|
|
|
| 6406487188 |
C_{\rm{Earth\ orbit}}, v_{\rm{Earth\ orbit}}, t_{\rm{Earth\ orbit}}
|
|
|
|
|
| 3046191961 |
v_{\rm{Earth\ orbit}} = \frac{C_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
|
|
|
|
|
| 3080027960 |
v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
|
|
|
|
|
| 7175416299 |
t_{\rm{Earth\ orbit}} = 1 {\rm{year}}
|
|
|
|
|
| 3219318145 |
\frac{365 {\rm days}}{1 {\rm year}} \frac{24 {\rm hours}}{1 {\rm day}} \frac{60 {\rm minutes}}{1 {\rm hour}} \frac{60 {\rm seconds}}{1 {\rm minute}}
|
|
|
|
|
| 8721295221 |
t_{\rm{Earth\ orbit}} = 3.16 10^7 {\rm{seconds}}
|
|
|
|
|
| 4593428198 |
v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{3.16\ 10^7 {\rm seconds}}
|
|
|
|
|
| 3472836147 |
r_{\rm{Earth\ orbit}} = 1.496\ 10^8 {\rm km}
|
|
|
|
|
| 6998364753 |
v_{\rm{Earth\ orbit}} = \frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}}
|
|
|
|
|
| 4180845508 |
v_{\rm{Earth\ orbit}} = 29.8 \frac{{\rm km}}{{\rm sec}}
|
|
|
|
|
| 4662369843 |
x' = \gamma (x - v t)
|
|
|
|
|
| 2983053062 |
x = \gamma (x' + v t')
|
|
|
|
|
| 3426941928 |
x = \gamma ( \gamma (x - v t) + v t' )
|
|
|
|
|
| 2096918413 |
x = \gamma ( \gamma x - \gamma v t + v t' )
|
|
|
|
|
| 7741202861 |
x = \gamma^2 x - \gamma^2 v t + \gamma v t'
|
|
|
|
|
| 7337056406 |
\gamma^2 x
|
|
|
|
|
| 4139999399 |
x - \gamma^2 x = - \gamma^2 v t + \gamma v t'
|
|
|
|
|
| 3495403335 |
x
|
|
|
|
|
| 9031609275 |
x (1 - \gamma^2 ) = - \gamma^2 v t + \gamma v t'
|
|
|
|
|
| 8014566709 |
\gamma^2 v t
|
|
|
|
|
| 9409776983 |
x (1 - \gamma^2 ) + \gamma^2 v t = \gamma v t'
|
|
|
|
|
| 2226340358 |
\gamma v
|
|
|
|
|
| 6417705759 |
\frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = \gamma v t'
|
|
|
|
|
| 8730201316 |
\frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t = t'
|
|
|
first term was multiplied by \gamma/\gamma
|
|
| 1974334644 |
\frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = t'
|
|
|
|
|
| 5148266645 |
t' = \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t
|
|
|
|
|
| 4287102261 |
x^2 + y^2 + z^2 = c^2 t^2
|
|
|
describes a spherical wavefront
|
|
| 1201689765 |
x'^2 + y'^2 + z'^2 = c^2 t'^2
|
|
|
describes a spherical wavefront for an observer in a moving frame of reference
|
|
| 7057864873 |
y' = y
|
|
|
frame of reference is moving only along x direction
|
|
| 8515803375 |
z' = z
|
|
|
frame of reference is moving only along x direction
|
|
| 6153463771 |
3
|
|
|
|
|
| 4746576736 |
asdf
|
|
|
|
|
| 1027270623 |
minga
|
|
|
|
|
| 6101803533 |
ming
|
|
|
|
|
| 9231495607 |
a = b
|
|
|
|
|
| 8664264194 |
b + 2
|
|
|
|
|
| 9805063945 |
\gamma^2 (x - v t)^2 + y^2 + z^2 = c^2 \gamma^2 \left( t + \frac{ 1 - \gamma^2 }{ \gamma^2 } \frac{x}{v} \right)^2
|
|
|
|
|
| 4057686137 |
C \to C_{\rm{Earth\ orbit}}
|
|
|
|
|
| 2346150725 |
r \to r_{\rm{Earth\ orbit}}
|
|
|
|
|
| 9753878784 |
v \to v_{\rm{Earth\ orbit}}
|
|
|
|
|
| 8135396036 |
t \to t_{\rm{Earth\ orbit}}
|
|
|
|
|
| 2773628333 |
\theta_1 \to \theta_{Brewster}
|
|
|
|
|
| 7154592211 |
\theta_2 \to \theta_{\rm refracted}
|
|
|
|
|
| 3749492596 |
E \to E_1
|
|
|
|
|
| 1258245373 |
E \to E_2
|
|
|
|
|
| 4147101187 |
KE \to KE_1
|
|
|
|
|
| 3809726424 |
PE \to PE_1
|
|
|
|
|
| 6383056612 |
KE \to KE_2
|
|
|
|
|
| 5075406409 |
PE \to PE_2
|
|
|
|
|
| 5398681503 |
v \to v_1
|
|
|
|
|
| 9305761407 |
v \to v_2
|
|
|
|
|
| 4218009993 |
x \to x_1
|
|
|
|
|
| 4188639044 |
x \to x_2
|
|
|
|
|
| 8173074178 |
x \to v_2
|
|
|
|
|
| 1025759423 |
y \to v_1
|
|
|
|
|
| 1935543849 |
\gamma^2 x^2 - \gamma^2 2 x v t + \gamma^2 v^2 t^2 + y^2 + z^2 = c^2 \gamma^2 \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x^2}{\gamma^2} + c^2 \gamma^2 2 t \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x}{\gamma} + c^2 \gamma^2 t^2
|
|
|
|
|
| 1586866563 |
\left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right) = t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)
|
|
|
|
|
| 3182633789 |
\gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1
|
|
|
|
|
| 1916173354 |
-\gamma^2 v^2 + c^2 \gamma^2 = c^2
|
|
|
|
|
| 2076171250 |
-\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} = 0
|
|
|
|
|
| 5284610349 |
\gamma^2
|
|
|
|
|
| 2417941373 |
- c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 - \gamma^2
|
|
|
|
|
| 5787469164 |
1 - \gamma^2
|
|
|
|
|
| 1639827492 |
- c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2} = 1
|
|
|
|
|
| 5669500954 |
v^2 \gamma^2
|
|
|
|
|
| 5763749235 |
-c^2 + c^2 \gamma^2 = v^2 \gamma^2
|
|
|
|
|
| 6408214498 |
c^2
|
|
|
|
|
| 2999795755 |
c^2 \gamma^2 = v^2 \gamma^2 + c^2
|
|
|
|
|
| 3412946408 |
v^2 \gamma^2
|
|
|
|
|
| 2542420160 |
c^2 \gamma^2 - v^2 \gamma^2 = c^2
|
|
|
|
|
| 7743841045 |
\gamma^2
|
|
|
|
|
| 7513513483 |
\gamma^2 (c^2 - v^2) = c^2
|
|
|
|
|
| 8571466509 |
c^2 - \gamma^2
|
|
|
|
|
| 7906112355 |
\gamma^2 = \frac{c^2}{c^2 - \gamma^2}
|
|
|
|
|
| 3060139639 |
1/2
|
|
|
|
|
| 1528310784 |
\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
|
|
|
|
|
| 8360117126 |
\gamma = \frac{-1}{\sqrt{1-\frac{v^2}{c^2}}}
|
|
|
not a physically valid result in this context
|
|
| 3366703541 |
a = \frac{v - v_0}{t}
|
|
|
acceleration is the average change in speed over a duration
|
|
| 7083390553 |
t
|
|
|
|
|
| 4748157455 |
a t = v - v_0
|
|
|
|
|
| 6417359412 |
v_0
|
|
|
|
|
| 4798787814 |
a t + v_0 = v
|
|
|
|
|
| 3462972452 |
v = v_0 + a t
|
|
|
|
|
| 3411994811 |
v_{\rm ave} = \frac{d}{t}
|
|
|
|
|
| 6175547907 |
v_{\rm ave} = \frac{v + v_0}{2}
|
|
|
|
|
| 9897284307 |
\frac{d}{t} = \frac{v + v_0}{2}
|
|
|
|
|
| 8865085668 |
t
|
|
|
|
|
| 8706092970 |
d = \left(\frac{v + v_0}{2}\right)t
|
|
|
|
|
| 7011114072 |
d = \frac{(v_0 + a t) + v_0}{2} t
|
|
|
|
|
| 1265150401 |
d = \frac{2 v_0 + a t}{2} t
|
|
|
|
|
| 9658195023 |
d = v_0 t + \frac{1}{2} a t^2
|
|
|
|
|
| 5799753649 |
2
|
|
|
|
|
| 7215099603 |
v^2 = v_0^2 + 2 a t v_0 + a^2 t^2
|
|
|
|
|
| 5144263777 |
v^2 = v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right)
|
|
|
|
|
| 7939765107 |
v^2 = v_0^2 + 2 a d
|
|
|
|
|
| 6729698807 |
v_0
|
|
|
|
|
| 9759901995 |
v - v_0 = a t
|
|
|
|
|
| 2242144313 |
a
|
|
|
|
|
| 1967582749 |
t = \frac{v - v_0}{a}
|
|
|
|
|
| 5733721198 |
d = \frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right)
|
|
|
|
|
| 5611024898 |
d = \frac{1}{2 a} (v^2 - v_0^2)
|
|
|
|
|
| 5542390646 |
2 a
|
|
|
|
|
| 8269198922 |
2 a d = v^2 - v_0^2
|
|
|
|
|
| 9070454719 |
v_0^2
|
|
|
|
|
| 4948763856 |
2 a d + v_0^2 = v^2
|
|
|
|
|
| 9645178657 |
a t
|
|
|
|
|
| 6457044853 |
v - a t = v_0
|
|
|
|
|
| 1259826355 |
d = (v - a t) t + \frac{1}{2} a t^2
|
|
|
|
|
| 4580545876 |
d = v t - a t^2 + \frac{1}{2} a t^2
|
|
|
|
|
| 6421241247 |
d = v t - \frac{1}{2} a t^2
|
|
|
|
|
| 4182362050 |
Z = |Z| \exp( i \theta )
|
|
|
Z \in \mathbb{C}
|
|
| 1928085940 |
Z^* = |Z| \exp( -i \theta )
|
|
|
|
|
| 9191880568 |
Z Z^* = |Z| |Z| \exp( -i \theta ) \exp( i \theta )
|
|
|
|
|
| 3350830826 |
Z Z^* = |Z|^2
|
|
|
|
|
| 8396997949 |
I = | A + B |^2
|
|
|
intensity of two waves traveling opposite directions on same path
|
|
| 4437214608 |
Z
|
|
|
|
|
| 5623794884 |
A + B
|
|
|
|
|
| 2236639474 |
(A + B)(A + B)^* = |A + B|^2
|
|
|
|
|
| 1020854560 |
I = (A + B)(A + B)^*
|
|
|
|
|
| 6306552185 |
I = (A + B)(A^* + B^*)
|
|
|
|
|
| 8065128065 |
I = A A^* + B B^* + A B^* + B A^*
|
|
|
|
|
| 9761485403 |
Z
|
|
|
|
|
| 8710504862 |
A
|
|
|
|
|
| 4075539836 |
A A^* = |A|^2
|
|
|
|
|
| 6529120965 |
B
|
|
|
|
|
| 1511199318 |
Z
|
|
|
|
|
| 7107090465 |
B B^* = |B|^2
|
|
|
|
|
| 5125940051 |
I = |A|^2 + B B^* + A B^* + B A^*
|
|
|
|
|
| 1525861537 |
I = |A|^2 + |B|^2 + A B^* + B A^*
|
|
|
|
|
| 1742775076 |
Z \to B
|
|
|
|
|
| 7607271250 |
\theta \to \phi
|
|
|
|
|
| 4192519596 |
B = |B| \exp(i \phi)
|
|
|
|
|
| 2064205392 |
A
|
|
|
|
|
| 1894894315 |
Z
|
|
|
|
|
| 1357848476 |
A = |A| \exp(i \theta)
|
|
|
|
|
| 6555185548 |
A^* = |A| \exp(-i \theta)
|
|
|
|
|
| 4504256452 |
B^* = |B| \exp(-i \phi)
|
|
|
|
|
| 7621705408 |
I = |A|^2 + |B|^2 + |A| |B| \exp(-i \theta) \exp(i \phi) + |A| |B| \exp(i \theta) \exp(-i \phi)
|
|
|
|
|
| 3085575328 |
I = |A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi))
|
|
|
|
|
| 4935235303 |
x
|
|
|
|
|
| 2293352649 |
\theta - \phi
|
|
|
|
|
| 3660957533 |
\cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
|
|
|
|
|
| 3967985562 |
2
|
|
|
|
|
| 2700934933 |
2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
|
|
|
|
|
| 8497631728 |
I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )
|
|
|
|
|
| 6774684564 |
\theta = \phi
|
|
|
for coherent waves
|
|
| 2099546947 |
I = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )
|
|
|
|
|
| 2719691582 |
|A| = |B|
|
|
|
in a loop
|
|
| 3257276368 |
I = |A|^2 + |A|^2 + |A| |A| 2
|
|
|
|
|
| 8283354808 |
I_{\rm coherent} = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )
|
|
|
|
|
| 8046208134 |
I_{\rm coherent} = |A|^2 + |A|^2 + |A| |A| 2
|
|
|
|
|
| 1172039918 |
I_{\rm coherent} = 4 |A|^2
|
|
|
|
|
| 8602221482 |
\langle \cos(\theta - \phi) \rangle = 0
|
|
|
incoherent light source
|
|
| 6240206408 |
I_{\rm incoherent} = |A|^2 + |B|^2
|
|
|
|
|
| 6529793063 |
I_{\rm incoherent} = |A|^2 + |A|^2
|
|
|
|
|
| 3060393541 |
I_{\rm incoherent} = 2|A|^2
|
|
|
|
|
| 6556875579 |
\frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2
|
|
|
|
|
| 8750500372 |
f
|
|
|
|
|
| 7543102525 |
g
|
|
|
|
|
| 8267133829 |
a = b
|
|
|
|
|
| 7968822469 |
c = d
|
|
|
|
|
| 3169580383 |
\vec{a} = \frac{d\vec{v}}{dt}
|
|
|
acceleration is the change in speed over a duration
|
|
| 8602512487 |
\vec{a} = a_x \hat{x} + a_y \hat{y}
|
|
|
decompose acceleration into two components
|
|
| 4158986868 |
a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt}
|
|
|
|
|
| 5349866551 |
\vec{v} = v_x \hat{x} + v_y \hat{y}
|
|
|
|
|
| 7729413831 |
a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right)
|
|
|
|
|
| 1819663717 |
a_x = \frac{d}{dt} v_x
|
|
|
|
|
| 8228733125 |
a_y = \frac{d}{dt} v_y
|
|
|
|
|
| 9707028061 |
a_x = 0
|
|
|
|
|
| 2741489181 |
a_y = -g
|
|
|
|
|
| 3270039798 |
2
|
|
|
|
|
| 8880467139 |
2
|
|
|
|
|
| 8750379055 |
0 = \frac{d}{dt} v_x
|
|
|
|
|
| 1977955751 |
-g = \frac{d}{dt} v_y
|
|
|
|
|
| 6672141531 |
dt
|
|
|
|
|
| 1702349646 |
-g dt = d v_y
|
|
|
|
|
| 8584698994 |
-g \int dt = \int d v_y
|
|
|
|
|
| 9973952056 |
-g t = v_y - v_{0, y}
|
|
|
|
|
| 4167526462 |
v_{0, y}
|
|
|
|
|
| 6572039835 |
-g t + v_{0, y} = v_y
|
|
|
|
|
| 7252338326 |
v_y = \frac{dy}{dt}
|
|
|
|
|
| 6204539227 |
-g t + v_{0, y} = \frac{dy}{dt}
|
|
|
|
|
| 1614343171 |
dt
|
|
|
|
|
| 8145337879 |
-g t dt + v_{0, y} dt = dy
|
|
|
|
|
| 8808860551 |
-g \int t dt + v_{0, y} \int dt = \int dy
|
|
|
|
|
| 2858549874 |
- \frac{1}{2} g t^2 + v_{0, y} t = y - y_0
|
|
|
|
|
| 6098638221 |
y_0
|
|
|
|
|
| 2461349007 |
- \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y
|
|
|
|
|
| 8717193282 |
dt
|
|
|
|
|
| 1166310428 |
0 dt = d v_x
|
|
|
|
|
| 2366691988 |
\int 0 dt = \int d v_x
|
|
|
|
|
| 1676472948 |
0 = v_x - v_{0, x}
|
|
|
|
|
| 1439089569 |
v_{0, x}
|
|
|
|
|
| 6134836751 |
v_{0, x} = v_x
|
|
|
|
|
| 8460820419 |
v_x = \frac{dx}{dt}
|
|
|
|
|
| 7455581657 |
v_{0, x} = \frac{dx}{dt}
|
|
|
|
|
| 8607458157 |
dt
|
|
|
|
|
| 1963253044 |
v_{0, x} dt = dx
|
|
|
|
|
| 3676159007 |
v_{0, x} \int dt = \int dx
|
|
|
|
|
| 9882526611 |
v_{0, x} t = x - x_0
|
|
|
|
|
| 3182907803 |
x_0
|
|
|
|
|
| 8486706976 |
v_{0, x} t + x_0 = x
|
|
|
|
|
| 1306360899 |
x = v_{0, x} t + x_0
|
|
|
|
|
| 7049769409 |
2
|
|
|
|
|
| 9341391925 |
\vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y}
|
|
|
|
|
| 6410818363 |
\theta
|
|
|
|
|
| 7391837535 |
\cos(\theta) = \frac{v_{0, x}}{v_0}
|
|
|
|
|
| 7376526845 |
\sin(\theta) = \frac{v_{0, y}}{v_0}
|
|
|
|
|
| 5868731041 |
v_0
|
|
|
|
|
| 6083821265 |
v_0 \cos(\theta) = v_{0, x}
|
|
|
|
|
| 5438722682 |
x = v_0 t \cos(\theta) + x_0
|
|
|
|
|
| 5620558729 |
v_0
|
|
|
|
|
| 8949329361 |
v_0 \sin(\theta) = v_{0, y}
|
|
|
|
|
| 1405465835 |
y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0
|
|
|
|
|
| 9862900242 |
y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0
|
|
|
|
|
| 6050070428 |
v_{0, x}
|
|
|
|
|
| 3274926090 |
t = \frac{x - x_0}{v_{0, x}}
|
|
|
|
|
| 7354529102 |
y = - \frac{1}{2} g \left( \frac{x - x_0}{v_{0, x}} \right)^2 + v_{0, y} \frac{x - x_0}{v_{0, x}} + y_0
|
|
|
|
|
| 8406170337 |
y \to y_f
|
|
|
|
|
| 2403773761 |
t \to t_f
|
|
|
|
|
| 5379546684 |
y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
|
|
|
|
|
| 5373931751 |
t = t_f
|
|
|
|
|
| 9112191201 |
y_f = 0
|
|
|
|
|
| 8198310977 |
0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
|
|
|
|
|
| 1650441634 |
y_0 = 0
|
|
|
define coordinate system such that initial height is at origin
|
|
| 1087417579 |
0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)
|
|
|
|
|
| 4829590294 |
t_f
|
|
|
|
|
| 2086924031 |
0 = - \frac{1}{2} g t_f + v_0 \sin(\theta)
|
|
|
|
|
| 6974054946 |
\frac{1}{2} g t_f
|
|
|
|
|
| 1191796961 |
\frac{1}{2} g t_f = v_0 \sin(\theta)
|
|
|
|
|
| 2510804451 |
2/g
|
|
|
|
|
| 4778077984 |
t_f = \frac{2 v_0 \sin(\theta)}{g}
|
|
|
|
|
| 3273630811 |
x \to x_f
|
|
|
|
|
| 6732786762 |
t \to t_f
|
|
|
|
|
| 3485125659 |
x_f = v_0 t_f \cos(\theta) + x_0
|
|
|
|
|
| 4370074654 |
t = t_f
|
|
|
|
|
| 2378095808 |
x_f = x_0 + d
|
|
|
|
|
| 4268085801 |
x_0 + d = v_0 t_f \cos(\theta) + x_0
|
|
|
|
|
| 8072682558 |
x_0
|
|
|
|
|
| 7233558441 |
d = v_0 t_f \cos(\theta)
|
|
|
|
|
| 2297105551 |
d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)
|
|
|
|
|
| 7587034465 |
\theta
|
|
|
|
|
| 7214442790 |
x
|
|
|
|
|
| 2519058903 |
\sin(2 \theta) = 2 \sin(\theta) \cos(\theta)
|
|
|
|
|
| 8922441655 |
d = \frac{v_0^2}{g} \sin(2 \theta)
|
|
|
|
|
| 5667870149 |
\theta
|
|
|
|
|
| 1541916015 |
\theta = \frac{\pi}{4}
|
|
|
|
|
| 3607070319 |
d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)
|
|
|
|
|
| 5353282496 |
d = \frac{v_0^2}{g}
|
|
|
|
|
| 0000040490 |
a^2
|
|
|
|
|
| 0000999900 |
b/(2a)
|
|
|
|
|
| 0001030901 |
\cos(x)
|
|
|
|
|
| 0001111111 |
(\sin(x))^2
|
|
|
|
|
| 0001209482 |
2 \pi
|
|
|
|
|
| 0001304952 |
\hbar
|
|
|
|
|
| 0001334112 |
W
|
|
|
|
|
| 0001921933 |
2 i
|
|
|
|
|
| 0002239424 |
2
|
|
|
|
|
| 0002338514 |
\vec{p}_{2}
|
|
|
|
|
| 0002342425 |
m/m
|
|
|
|
|
| 0002367209 |
1/2
|
|
|
|
|
| 0002393922 |
x
|
|
|
|
|
| 0002424922 |
a
|
|
|
|
|
| 0002436656 |
i \hbar
|
|
|
|
|
| 0002449291 |
b/(2a)
|
|
|
|
|
| 0002838490 |
b/(2a)
|
|
|
|
|
| 0002919191 |
\sin(-x)
|
|
|
|
|
| 0002929944 |
1/2
|
|
|
|
|
| 0002940021 |
2 \pi
|
|
|
|
|
| 0003232242 |
t
|
|
|
|
|
| 0003413423 |
\cos(-x)
|
|
|
|
|
| 0003747849 |
-1
|
|
|
|
|
| 0003838111 |
2
|
|
|
|
|
| 0003919391 |
x
|
|
|
|
|
| 0003949052 |
-x
|
|
|
|
|
| 0003949921 |
\hbar
|
|
|
|
|
| 0003954314 |
dx
|
|
|
|
|
| 0003981813 |
-\sin(x)
|
|
|
|
|
| 0004089571 |
2x
|
|
|
|
|
| 0004264724 |
y
|
|
|
|
|
| 0004307451 |
(b/(2a))^2
|
|
|
|
|
| 0004582412 |
x
|
|
|
|
|
| 0004829194 |
2
|
|
|
|
|
| 0004831494 |
a
|
|
|
|
|
| 0004849392 |
x
|
|
|
|
|
| 0004858592 |
h
|
|
|
|
|
| 0004934845 |
x
|
|
|
|
|
| 0004948585 |
a
|
|
|
|
|
| 0005395034 |
a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
|
|
|
|
|
| 0005626421 |
t
|
|
|
|
|
| 0005749291 |
f
|
|
|
|
|
| 0005938585 |
\frac{-\hbar^2}{2m}
|
|
|
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| 0006466214 |
(\sin(x))^2
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| 0006544644 |
t
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| 0006563727 |
x
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| 0006644853 |
c/a
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| 0006656532 |
e
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| 0007471778 |
2(\sin(x))^2
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| 0007563791 |
i
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| 0007636749 |
x
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| 0007894942 |
(\sin(x))^2
|
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| 0008837284 |
T
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| 0008842811 |
\cos(2x)
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| 0009458842 |
\psi(x)
|
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| 0009484724 |
\frac{n \pi}{W}x
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| 0009485857 |
a^2\frac{2}{W}
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| 0009485858 |
\frac{2n\pi}{W}
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| 0009492929 |
v du
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| 0009587738 |
\psi
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| 0009594995 |
1/2
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| 0009877781 |
y
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| 0203024440 |
1 = \int_0^W a \sin(\frac{n \pi}{W} x) \psi(x)^* dx
|
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|
|
| 0404050504 |
\lambda = \frac{v}{f}
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| 0439492440 |
\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)\right|_0^W
|
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| 0934990943 |
k = \frac{2 \pi}{v T}
|
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| 0948572140 |
\int \cos(a x) dx = \frac{1}{a}\sin(a x)
|
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| 1010393913 |
\langle \psi| \hat{A}^+ |\psi \rangle = \langle a \rangle^*
|
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| 1010393944 |
x = \langle\psi_{\alpha}| a_{\beta} |\psi_{\beta} \rangle
|
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| 1010923823 |
k W = n \pi
|
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| 1020010291 |
0 = a \sin(k W)
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| 1020394900 |
p = h/\lambda
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| 1020394902 |
E = h f
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| 1029039903 |
p = m v
|
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| 1029039904 |
p^2 = m^2 v^2
|
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| 1158485859 |
\frac{-\hbar^2}{2m} \nabla^2 = {\cal H}
|
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| 1202310110 |
\frac{1}{a^2} = \int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
|
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| 1202312210 |
\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
|
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| 1203938249 |
a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle = a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
|
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| 1248277773 |
\cos(2x) = 1-2(\sin(x))^2
|
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| 1293913110 |
0 = b
|
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| 1293923844 |
\lambda = v T
|
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| 1314464131 |
\vec{\nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
|
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| 1314864131 |
\vec{\nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E}
|
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| 1395858355 |
x = \langle \psi_{\alpha}| a_{\alpha} |\psi_{\beta}\rangle
|
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| 1492811142 |
f = x - d
|
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| 1492842000 |
\nabla \vec{x} = f(y)
|
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| 1636453295 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = - \nabla^2 \vec{E}
|
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|
|
| 1638282134 |
\vec{p}_{before} = \vec{p}_{after}
|
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|
|
|
| 1648958381 |
\nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{\nabla}\psi(\vec{r},t)\right)
|
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|
|
|
| 1857710291 |
0 = a \sin(n \pi)
|
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|
| 1858578388 |
\nabla^2 E(\vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E(\vec{r})\exp(i \omega t)
|
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|
|
|
| 1858772113 |
k = \frac{n \pi}{W}
|
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|
|
| 1934748140 |
\int |\psi(x)|^2 dx = 1
|
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|
|
|
| 2029293929 |
\nabla^2 E(\vec{r})\exp(i \omega t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r})\exp(i \omega t)
|
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|
|
|
| 2103023049 |
\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right)
|
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|
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|
| 2113211456 |
f = 1/T
|
|
|
|
|
| 2123139121 |
-\exp(-i x) = -\cos(x)+i \sin(x)
|
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|
|
|
| 2131616531 |
T f = 1
|
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|
|
|
| 2258485859 |
{\cal H} \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
|
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|
|
| 2394240499 |
x = a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle
|
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|
|
|
| 2394853829 |
\exp(-i x) = \cos(-x)+i \sin(-x)
|
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|
| 2394935831 |
( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
|
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|
|
| 2394935835 |
\left(\langle\psi| \hat{A} |\psi\rangle \right)^+ = \left(\langle a \rangle\right)^+
|
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|
|
|
| 2395958385 |
\nabla^2 \psi\left(\vec{r},t)\right) = \frac{-p^2}{\hbar} \psi(\vec{r},t)
|
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|
|
|
| 2404934990 |
\langle x^2\rangle -2\langle x \rangle\langle x \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
|
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|
|
|
| 2494533900 |
\langle x^2\rangle -\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
|
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|
|
|
| 2569154141 |
\vec{\nabla} \times \frac{\partial}{\partial t}\vec{H} = \epsilon_0 \frac{\partial^2 }{\partial t^2}\vec{E}
|
|
|
|
|
| 2648958382 |
\nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p}\psi(\vec{r},t)\right)
|
|
|
|
|
| 2848934890 |
\langle a \rangle^* = \langle a \rangle
|
|
|
|
|
| 2944838499 |
\psi(x) = a \sin(\frac{n \pi}{W} x)
|
|
|
|
|
| 3121234211 |
\frac{k}{2\pi} = \lambda
|
|
|
|
|
| 3121234212 |
p = \frac{h k}{2\pi}
|
|
|
|
|
| 3121513111 |
k = \frac{2 \pi}{\lambda}
|
|
|
|
|
| 3131111133 |
T = 1 / f
|
|
|
|
|
| 3131211131 |
\omega = 2 \pi f
|
|
|
|
|
| 3132131132 |
\omega = \frac{2\pi}{T}
|
|
|
|
|
| 3147472131 |
\frac{\omega}{2 \pi} = f
|
|
|
|
|
| 3285732911 |
(\cos(x))^2 = 1-(\sin(x))^2
|
|
|
|
|
| 3485475729 |
\nabla^2 E(\vec{r}) = - \frac{\omega^2}{c^2} E(\vec{r})
|
|
|
|
|
| 3585845894 |
\langle \left(x-\langle x \rangle\right)^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 3829492824 |
\frac{1}{2}\left(\exp(i x)+\exp(-i x)\right) = \cos(x)
|
|
|
|
|
| 3924948349 |
a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
|
|
|
|
|
| 3942849294 |
\exp(i x)-\exp(-i x) = 2 i \sin(x)
|
|
|
|
|
| 3943939590 |
x = a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle
|
|
|
|
|
| 3947269979 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
|
|
|
|
|
| 3948571256 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi(\vec{r},t)
|
|
|
|
|
| 3948572230 |
\vec{\nabla}\psi(\vec{r},t) = \psi_0 \vec{\nabla}\exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
|
|
|
|
|
| 3948574224 |
\psi(\vec{r},t) = \psi_0 \exp\left(i\left(\vec{k}\cdot\vec{r} - \omega t \right) \right)
|
|
|
|
|
| 3948574226 |
\psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right)
|
|
|
|
|
| 3948574228 |
\psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
|
|
|
|
|
| 3948574230 |
\psi(\vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
|
|
|
|
|
| 3948574233 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
|
|
|
|
|
| 3948574235 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
|
|
|
|
|
| 3951205425 |
\vec{p}_{after} = \vec{p}_{1}
|
|
|
|
|
| 4147472132 |
E = \frac{h \omega}{2 \pi}
|
|
|
|
|
| 4298359835 |
E = \frac{1}{2}m v^2
|
|
|
|
|
| 4298359845 |
E = \frac{1}{2m}m^2 v^2
|
|
|
|
|
| 4298359851 |
E = \frac{p^2}{2m}
|
|
|
|
|
| 4341171256 |
i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{p^2}{2 m} \psi(\vec{r},t)
|
|
|
|
|
| 4348571256 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi(\vec{r},t)
|
|
|
|
|
| 4394958389 |
\vec{\nabla}\cdot \left(\vec{\nabla}\psi(\vec{r},t)\right) = \frac{i}{\hbar} \vec{\nabla}\cdot\left( \vec{p} \psi(\vec{r},t)\right)
|
|
|
|
|
| 4585828572 |
\epsilon_0 \mu_0 = \frac{1}{c^2}
|
|
|
|
|
| 4585932229 |
\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
|
|
|
|
|
| 4598294821 |
\exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
|
|
|
|
|
| 4638429483 |
\exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))
|
|
|
|
|
| 4742644828 |
\exp(i x)+\exp(-i x) = 2 \cos(x)
|
|
|
|
|
| 4827492911 |
\cos(2x)+(\sin(x))^2 = 1-(\sin(x))^2
|
|
|
|
|
| 4838429483 |
\exp(2 i x) = \cos(2 x)+i \sin(2 x)
|
|
|
|
|
| 4843995999 |
\frac{1}{2 i}\left(\exp(i x)-\exp(-i x)\right) = \sin(x)
|
|
|
|
|
| 4857472413 |
1 = \int \psi(x)\psi(x)^* dx
|
|
|
|
|
| 4857475848 |
\frac{1}{a^2} = \frac{W}{2}
|
|
|
|
|
| 4858429483 |
\exp(i x)\exp(i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))
|
|
|
|
|
| 4923339482 |
i x = \log(y)
|
|
|
|
|
| 4928239482 |
\log(y) = i x
|
|
|
|
|
| 4928923942 |
a = b
|
|
|
|
|
| 4929218492 |
a+b = c
|
|
|
|
|
| 4929482992 |
b = 2
|
|
|
|
|
| 4938429482 |
\exp(-i x) = \cos(x)+i \sin(-x)
|
|
|
|
|
| 4938429483 |
\exp(i x) = \cos(x)+i \sin(x)
|
|
|
|
|
| 4938429484 |
\exp(-i x) = \cos(x)-i \sin(x)
|
|
|
|
|
| 4943571230 |
\vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
|
|
|
|
|
| 4948934890 |
\langle \psi| \hat{A} |\psi\rangle = \langle a \rangle^*
|
|
|
|
|
| 4949359835 |
\langle x^2\rangle -2\langle x^2 \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 4954839242 |
\cos(2x)+i\sin(2x) = (\cos(x)+i \sin(x))(\cos(x)+i \sin(x))
|
|
|
|
|
| 4978429483 |
\exp(-i x) = \cos(-x)+i \sin(-x)
|
|
|
|
|
| 4984892984 |
a+2 = c
|
|
|
|
|
| 4985825552 |
\nabla^2 E(\vec{r})\exp(i \omega t) = i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 5530148480 |
\vec{p}_{1}-\vec{p}_{2} = \vec{p}_{electron}
|
|
|
|
|
| 5727578862 |
\frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)
|
|
|
|
|
| 5832984291 |
(\sin(x))^2 + (\cos(x))^2 = 1
|
|
|
|
|
| 5857434758 |
\int a dx = a x
|
|
|
|
|
| 5868688585 |
\frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = \frac{p^2}{2m} \psi(\vec{r},t)
|
|
|
|
|
| 5900595848 |
k = \frac{\omega}{v}
|
|
|
|
|
| 5928285821 |
x^2 + 2x(b/(2a)) + (b/(2a))^2 = (x+(b/(2a)))^2
|
|
|
|
|
| 5928292841 |
x^2 + (b/a)x + (b/(2a))^2 = -c/a + (b/(2a))^2
|
|
|
|
|
| 5938459282 |
x^2 + (b/a)x = -c/a
|
|
|
|
|
| 5958392859 |
x^2 + (b/a)x+(c/a) = 0
|
|
|
|
|
| 5959282914 |
x^2 + x(b/a) + (b/(2a))^2 = (x+(b/(2a)))^2
|
|
|
|
|
| 5982958248 |
x = -\sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
|
|
|
|
|
| 5982958249 |
x+(b/(2a)) = -\sqrt{(b/(2a))^2 - (c/a)}
|
|
|
|
|
| 5985371230 |
\vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi(\vec{r},t)
|
|
|
|
|
| 6742123016 |
\vec{p}_{electron}\cdot\vec{p}_{electron} = (\vec{p}_{1}\cdot\vec{p}_{1})+(\vec{p}_{2}\cdot\vec{p}_{2})-2(\vec{p}_{1}\cdot\vec{p}_{2})
|
|
|
|
|
| 7466829492 |
\vec{\nabla} \cdot \vec{E} = 0
|
|
|
|
|
| 7564894985 |
\int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)
|
|
|
|
|
| 7572664728 |
\cos(2x)+2(\sin(x))^2 = 1
|
|
|
|
|
| 7575738420 |
\left(\sin\left(\frac{n \pi}{W}x\right)\right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2}
|
|
|
|
|
| 7575859295 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859300 |
\epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{\nabla} \times \vec{E} )_k = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859302 |
\epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859304 |
\epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h}
|
|
|
|
|
| 7575859306 |
\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859308 |
\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right) = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859310 |
\hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859312 |
\vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7917051060 |
\vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2}
|
|
|
|
|
| 8139187332 |
\vec{p}_{1} = \vec{p}_{2}+\vec{p}_{electron}
|
|
|
|
|
| 8257621077 |
\vec{p}_{before} = \vec{p}_{1}
|
|
|
|
|
| 8311458118 |
\vec{p}_{after} = \vec{p}_{2}+\vec{p}_{electron}
|
|
|
|
|
| 8399484849 |
\langle x^2 -2x\langle x \rangle+\langle x \rangle^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 8484544728 |
-a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x)
|
|
|
|
|
| 8485757728 |
a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(kx)
|
|
|
|
|
| 8485867742 |
\frac{2}{W} = a^2
|
|
|
|
|
| 8489593958 |
d(u v) = u dv + v du
|
|
|
|
|
| 8489593960 |
d(u v) - v du = u dv
|
|
|
|
|
| 8489593962 |
u dv = d(u v) - v du
|
|
|
|
|
| 8489593964 |
\int u dv = u v - \int v du
|
|
|
|
|
| 8494839423 |
\nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
|
|
|
|
|
| 8572657110 |
1 = \int |\psi(x)|^2 dx
|
|
|
|
|
| 8572852424 |
\vec{E} = E(\vec{r},t)
|
|
|
|
|
| 8575746378 |
\int \frac{1}{2} dx = \frac{1}{2} x
|
|
|
|
|
| 8575748999 |
\frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x)\right) = -k^2 \left(a \sin(kx) + b \cos(kx)\right)
|
|
|
|
|
| 8576785890 |
1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
|
|
|
|
|
| 8577275751 |
0 = a \sin(0) + b\cos(0)
|
|
|
|
|
| 8582885111 |
\psi(x) = a \sin(kx) + b \cos(kx)
|
|
|
|
|
| 8582954722 |
x^2 + 2xh + h^2 = (x+h)^2
|
|
|
|
|
| 8849289982 |
\psi(x)^* = a \sin(\frac{n \pi}{W} x)
|
|
|
|
|
| 8889444440 |
1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right)\right)^2 dx
|
|
|
|
|
| 9059289981 |
\psi(x) = a \sin(k x)
|
|
|
|
|
| 9285928292 |
ax^2 + bx + c = 0
|
|
|
|
|
| 9291999979 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0\vec{\nabla} \times \frac{\partial \vec{H}}{\partial t}
|
|
|
|
|
| 9294858532 |
\hat{A}^+ = \hat{A}
|
|
|
|
|
| 9385938295 |
(x+(b/(2a)))^2 = -(c/a) + (b/(2a))^2
|
|
|
|
|
| 9393939991 |
\psi(x) = -\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
|
|
|
|
|
| 9393939992 |
\psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
|
|
|
|
|
| 9394939493 |
\nabla^2 E(\vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r},t)
|
|
|
|
|
| 9429829482 |
\frac{d}{dx} y = -\sin(x) + i\cos(x)
|
|
|
|
|
| 9482113948 |
\frac{dy}{y} = i dx
|
|
|
|
|
| 9482438243 |
(\cos(x))^2 = \cos(2x)+(\sin(x))^2
|
|
|
|
|
| 9482923849 |
\exp(i x) = y
|
|
|
|
|
| 9482928242 |
\cos(2x) = (\cos(x))^2 - (\sin(x))^2
|
|
|
|
|
| 9482928243 |
\cos(2x)+(\sin(x))^2 = (\cos(x))^2
|
|
|
|
|
| 9482943948 |
\log(y) = i dx
|
|
|
|
|
| 9482948292 |
b = c
|
|
|
|
|
| 9482984922 |
\frac{d}{dx} y = (i\sin(x) + \cos(x)) i
|
|
|
|
|
| 9483928192 |
\cos(2x)+i\sin(2x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
|
|
|
|
|
| 9485384858 |
\nabla^2 E(\vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 9485747245 |
\sqrt{\frac{2}{W}} = a
|
|
|
|
|
| 9485747246 |
-\sqrt{\frac{2}{W}} = a
|
|
|
|
|
| 9492920340 |
y = \cos(x)+i \sin(x)
|
|
|
|
|
| 9494829190 |
k
|
|
|
|
|
| 9495857278 |
\psi(x=W) = 0
|
|
|
|
|
| 9499428242 |
E(\vec{r},t) = E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 9499959299 |
a + k = b + k
|
|
|
|
|
| 9582958293 |
x = \sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
|
|
|
|
|
| 9582958294 |
x+(b/(2a)) = \sqrt{(b/(2a))^2 - (c/a)}
|
|
|
|
|
| 9584821911 |
c + d = a
|
|
|
|
|
| 9585727710 |
\psi(x=0) = 0
|
|
|
|
|
| 9596004948 |
x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle
|
|
|
|
|
| 9848292229 |
dy = y i dx
|
|
|
|
|
| 9848294829 |
\frac{d}{dx} y = y i
|
|
|
|
|
| 9858028950 |
\frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
|
|
|
|
|
| 9889984281 |
2(\sin(x))^2 = 1-\cos(2x)
|
|
|
|
|
| 9919999981 |
\rho = 0
|
|
|
|
|
| 9958485859 |
\frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
|
|
|
|
|
| 9988949211 |
(\sin(x))^2 = \frac{1-\cos(2x)}{2}
|
|
|
|
|
| 9991999979 |
\vec{\nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t}
|
|
|
|
|
| 9999998870 |
\frac{\vec{p}}{\hbar} = \vec{k}
|
|
|
|
|
| 9999999870 |
\frac{p}{\hbar} = k
|
|
|
|
|
| 9999999950 |
\beta = 1/(k_{Boltzmann}T)
|
|
|
|
|
| 9999999951 |
\langle x | k \rangle = \frac{\exp(i k x)}{\sqrt{2\pi}}
|
|
|
|
|
| 9999999952 |
f(x) \delta(x-a) = f(a)
|
|
|
|
|
| 9999999953 |
\int_{-\infty}^{\infty} \delta(x) dx = 1
|
|
|
|
|
| 9999999954 |
c = 1/\sqrt{\epsilon_0 \mu_0}
|
|
|
|
|
| 9999999955 |
\vec{E} = -\vec{\nabla} \Psi
|
|
|
|
|
| 9999999956 |
\vec{F} = \frac{\partial}{\partial t}\vec{p}
|
|
|
|
|
| 9999999957 |
\vec{F} = -\vec{\nabla} V
|
|
|
|
|
| 9999999960 |
\hbar = h/(2 \pi)
|
|
|
|
|
| 9999999961 |
\frac{E}{\hbar} = \omega
|
|
|
|
|
| 9999999962 |
p = \hbar k
|
|
|
|
|
| 9999999963 |
\lambda = h/p
|
|
|
|
|
| 9999999964 |
\omega = c k
|
|
|
|
|
| 9999999965 |
E = \omega \hbar
|
|
|
|
|
| 9999999966 |
\vec{L} = \vec{r}\times\vec{p}
|
|
|
|
|
| 9999999967 |
\vec{S} = \frac{1}{\mu_0} \vec{E}\times \vec{B}
|
|
|
|
|
| 9999999968 |
x = \frac{-b-\sqrt{b^2-4ac}}{2a}
|
|
|
|
|
| 9999999969 |
x = \frac{-b+\sqrt{b^2-4ac}}{2a}
|
|
|
|
|
| 9999999970 |
\eta_1 \sin\theta_1 = \eta_2 \sin\theta_2
|
|
|
|
|
| 9999999971 |
{\cal H} = \frac{p^2}{2m}+V
|
|
|
|
|
| 9999999972 |
{\cal H} |\psi\rangle = E |\psi\rangle
|
|
|
|
|
| 9999999973 |
\left( \Delta A \right)^2 = \langle A^2 \rangle - \langle A \rangle^2
|
|
|
|
|
| 9999999974 |
\langle \psi| \hat{A} |\psi\rangle = \int_{-\infty}^{\infty} \psi^* A \psi dx
|
|
|
|
|
| 9999999975 |
\langle \psi| \hat{A} |\psi\rangle = \langle a \rangle
|
|
|
|
|
| 9999999976 |
\hat{p} = -i \hbar \frac{\partial }{\partial x}
|
|
|
|
|
| 9999999977 |
[\hat{x},\hat{p}] = i \hbar
|
|
|
|
|
| 9999999978 |
\vec{\nabla} \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial E}{\partial t}
|
|
|
|
|
| 9999999979 |
\vec{\nabla} \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}
|
|
|
|
|
| 9999999980 |
\vec{\nabla} \cdot \vec{B} = 0
|
|
|
|
|
| 9999999981 |
\vec{\nabla} \cdot \vec{E} = \rho/\epsilon_0
|
|
|
|
|
| 9999999982 |
V = I R + Q/C + L \frac{\partial I}{\partial t}
|
|
|
|
|
| 9999999983 |
C = V A/d
|
|
|
|
|
| 9999999984 |
Q = C V
|
|
|
|
|
| 9999999985 |
V = I R
|
|
|
|
|
| 9999999986 |
\left[\right]\left[\right] = \left[\right]
|
|
|
|
|
| 9999999987 |
1 atmosphere = 101.325 Pascal
|
|
|
|
|
| 9999999988 |
1 atmosphere = 14.7 pounds/(inch^2)
|
|
|
|
|
| 9999999989 |
mass_{electron} = 511000 electronVolts/(q^2)
|
|
|
|
|
| 9999999990 |
Tesla = 10000*Gauss
|
|
|
|
|
| 9999999991 |
Pascal = Newton / (meter^2)
|
|
|
|
|
| 9999999992 |
Tesla = Newton / (Ampere*meter)
|
|
|
|
|
| 9999999993 |
Farad = Coulumb / Volt
|
|
|
|
|
| 9999999995 |
Volt = Joule / Coulumb
|
|
|
|
|
| 9999999996 |
Coulomb = Ampere / second
|
|
|
|
|
| 9999999997 |
Watt = Joule / second
|
|
|
|
|
| 9999999998 |
Joule = Newton*meter
|
|
|
|
|
| 9999999999 |
Newton = kilogram*meter/(second^2)
|
|
|
|
|
| 4961662865 |
x
|
|
|
|
|
| 9110536742 |
2 x
|
|
|
|
|
| 8483686863 |
\sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x)\right)
|
|
|
|
|
| 3470587782 |
\sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
|
|
|
|
|
| 8642992037 |
2
|
|
|
|
|
| 9894826550 |
2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \left(\exp(i x)+\exp(-i x)\right)
|
|
|
|
|
| 4257214632 |
\frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)
|
|
|
|
|
| 8699789241 |
2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)
|
|
|
|
|
| 9180861128 |
2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)
|
|
|
|
|
| 2405307372 |
\sin(2 x) = 2 \sin(x) \cos(x)
|
|
|
|
|
| 3268645065 |
x
|
|
|
|
|
| 9350663581 |
\pi
|
|
|
|
|
| 8332931442 |
\exp(i \pi) = \cos(\pi)+i \sin(\pi)
|
|
|
|
|
| 6885625907 |
\exp(i \pi) = -1 + i 0
|
|
|
|
|
| 3331824625 |
\exp(i \pi) = -1
|
|
|
|
|
| 4901237716 |
1
|
|
|
|
|
| 2501591100 |
\exp(i \pi) + 1 = 0
|
|
|
|
|
| 5136652623 |
E = KE + PE
|
|
mechanical energy is the sum of the potential plus kinetic energies |
|
|
| 7915321076 |
E, KE, PE
|
|
|
|
|
| 3192374582 |
E_2, KE_2, PE_2
|
|
|
|
|
| 7875206161 |
E_2 = KE_2 + PE_2
|
|
|
|
|
| 4229092186 |
E, KE, PE
|
|
|
|
|
| 1490821948 |
E_1, KE_1, PE_1
|
|
|
|
|
| 4303372136 |
E_1 = KE_1 + PE_1
|
|
|
|
|
| 5514556106 |
E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1)
|
|
|
|
|
| 8357234146 |
KE = \frac{1}{2} m v^2
|
|
kinetic energy |
Kinetic_energy
|
|
| 3658066792 |
KE_2, v_2
|
|
|
|
|
| 8082557839 |
KE, v
|
|
|
|
|
| 7676652285 |
KE_2 = \frac{1}{2} m v_2^2
|
|
|
|
|
| 2790304597 |
KE_1, v_1
|
|
|
|
|
| 9880327189 |
KE, v
|
|
|
|
|
| 4928007622 |
KE_1 = \frac{1}{2} m v_1^2
|
|
|
|
|
| 5733146966 |
KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right)
|
|
|
|
|
| 6554292307 |
t
|
|
|
|
|
| 4270680309 |
\frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t}
|
|
|
|
|
| 5781981178 |
x^2 - y^2 = (x+y)(x-y)
|
|
difference of squares |
Difference_of_two_squares
|
|
| 5946759357 |
v_2, v_1
|
|
|
|
|
| 6675701064 |
x, y
|
|
|
|
|
| 4648451961 |
v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1)
|
|
|
|
|
| 9356924046 |
\frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t}
|
|
|
|
|
| 9397152918 |
v = \frac{v_1 + v_2}{2}
|
|
average velocity |
|
|
| 2857430695 |
a = \frac{v_2 - v_1}{t}
|
|
acceleration |
|
|
| 7735737409 |
\frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t}
|
|
|
|
|
| 4784793837 |
\frac{KE_2 - KE_1}{t} = m v a
|
|
|
|
|
| 5345738321 |
F = m a
|
|
Newton's second law of motion |
Newton%27s_laws_of_motion#Newton's_second_law
|
|
| 2186083170 |
\frac{KE_2 - KE_1}{t} = v F
|
|
|
|
|
| 6715248283 |
PE = -F x
|
|
potential energy |
Potential_energy
|
|
| 3852078149 |
PE_2, x_2
|
|
|
|
|
| 7388921188 |
PE, x
|
|
|
|
|
| 2431507955 |
PE_2 = -F x_2
|
|
|
|
|
| 3412641898 |
PE_1, x_1
|
|
|
|
|
| 9937169749 |
PE, x
|
|
|
|
|
| 4669290568 |
PE_1 = -F x_1
|
|
|
|
|
| 7734996511 |
PE_2 - PE_1 = -F ( x_2 - x_1 )
|
|
|
|
|
| 2016063530 |
t
|
|
|
|
|
| 7267155233 |
\frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right)
|
|
|
|
|
| 9337785146 |
v = \frac{x_2 - x_1}{t}
|
|
average velocity |
|
|
| 4872970974 |
\frac{PE_2 - PE_1}{t} = -F v
|
|
|
|
|
| 2081689540 |
t
|
|
|
|
|
| 2770069250 |
\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t}
|
|
|
|
|
| 3591237106 |
\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v
|
|
|
|
|
| 1772416655 |
\frac{E_2 - E_1}{t} = v F - F v
|
|
|
|
|
| 1809909100 |
\frac{E_2 - E_1}{t} = 0
|
|
|
|
|
| 5778176146 |
t
|
|
|
|
|
| 3806977900 |
E_2 - E_1 = 0
|
|
|
|
|
| 5960438249 |
E_1
|
|
|
|
|
| 8558338742 |
E_2 = E_1
|
|
|
|
|
| 8945218208 |
\theta_{Brewster} + \theta_{refracted} = 90^{\circ}
|
|
|
based on figure 34-27 on page 824 in 2001_HRW
|
|
| 9025853427 |
\theta_{Brewster}
|
|
|
|
|
| 1310571337 |
\theta_{refracted} = 90^{\circ} - \theta_{Brewster}
|
|
|
|
|
| 6450985774 |
n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 )
|
|
Law of Refraction |
eq 34-44 on page 819 in 2001_HRW
|
|
| 1779497048 |
\theta_{Brewster}, \theta_{refraction}
|
|
|
|
|
| 7322344455 |
\theta_1, \theta_2
|
|
|
|
|
| 2575937347 |
n_1 \sin( \theta_{Brewster} ) = n_2 \sin( \theta_{refraction} )
|
|
|
|
|
| 7696214507 |
n_1 \sin( \theta_{Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{Brewster} )
|
|
|
|
|
| 8588429722 |
\sin( 90^{\circ} - x ) = \cos( x )
|
|
|
|
|
| 7375348852 |
\theta_{Brewster}
|
|
|
|
|
| 1512581563 |
x
|
|
|
|
|
| 6831637424 |
\sin( 90^{\circ} - \theta_{Brewster} ) = \cos( \theta_{Brewster} )
|
|
|
|
|
| 3061811650 |
n_1 \sin( \theta_{Brewster} ) = n_2 \cos( \theta_{Brewster} )
|
|
|
|
|
| 7857757625 |
n_1
|
|
|
|
|
| 9756089533 |
\sin( \theta_{Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{Brewster} )
|
|
|
|
|
| 5632428182 |
\cos( \theta_{Brewster} )
|
|
|
|
|
| 2768857871 |
\frac{\sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )} = \frac{n_2}{n_1}
|
|
|
|
|
| 4968680693 |
\tan( x ) = \frac{ \sin( x )}{\cos( x )}
|
|
|
|
|
| 7321695558 |
\theta_{Brewster}
|
|
|
|
|
| 9906920183 |
x
|
|
|
|
|
| 4501377629 |
\tan( \theta_{Brewster} ) = \frac{ \sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )}
|
|
|
|
|
| 3417126140 |
\tan( \theta_{Brewster} ) = \frac{ n_2 }{ n_1 }
|
|
|
|
|
| 5453995431 |
\arctan{ x }
|
|
|
|
|
| 6023986360 |
x
|
|
|
|
|
| 8495187962 |
\theta_{Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }
|
|
|
|
|
| 9040079362 |
f
|
|
|
|
|
| 9565166889 |
T
|
|
|
|
|
| 5426308937 |
v = \frac{d}{t}
|
|
|
|
|
| 7476820482 |
C
|
|
|
|
|
| 1277713901 |
d
|
|
|
|
|
| 6946088325 |
v = \frac{C}{t}
|
|
|
|
|
| 6785303857 |
C = 2 \pi r
|
|
|
|
|
| 4816195267 |
C, r
|
|
|
|
|
| 2113447367 |
C_{\rm{Earth\ orbit}}, r_{\rm{Earth\ orbit}}
|
|
|
|
|
| 6348260313 |
C_{\rm{Earth\ orbit}} = 2 \pi r_{\rm{Earth\ orbit}}
|
|
|
|
|
| 3847777611 |
C, v, t
|
|
|
|
|
| 6406487188 |
C_{\rm{Earth\ orbit}}, v_{\rm{Earth\ orbit}}, t_{\rm{Earth\ orbit}}
|
|
|
|
|
| 3046191961 |
v_{\rm{Earth\ orbit}} = \frac{C_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
|
|
|
|
|
| 3080027960 |
v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
|
|
|
|
|
| 7175416299 |
t_{\rm{Earth\ orbit}} = 1 {\rm{year}}
|
|
|
|
|
| 3219318145 |
\frac{365 {\rm days}}{1 {\rm year}} \frac{24 {\rm hours}}{1 {\rm day}} \frac{60 {\rm minutes}}{1 {\rm hour}} \frac{60 {\rm seconds}}{1 {\rm minute}}
|
|
|
|
|
| 8721295221 |
t_{\rm{Earth\ orbit}} = 3.16 10^7 {\rm{seconds}}
|
|
|
|
|
| 4593428198 |
v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{3.16\ 10^7 {\rm seconds}}
|
|
|
|
|
| 3472836147 |
r_{\rm{Earth\ orbit}} = 1.496\ 10^8 {\rm km}
|
|
|
|
|
| 6998364753 |
v_{\rm{Earth\ orbit}} = \frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}}
|
|
|
|
|
| 4180845508 |
v_{\rm{Earth\ orbit}} = 29.8 \frac{{\rm km}}{{\rm sec}}
|
|
|
|
|
| 4662369843 |
x' = \gamma (x - v t)
|
|
|
|
|
| 2983053062 |
x = \gamma (x' + v t')
|
|
|
|
|
| 3426941928 |
x = \gamma ( \gamma (x - v t) + v t' )
|
|
|
|
|
| 2096918413 |
x = \gamma ( \gamma x - \gamma v t + v t' )
|
|
|
|
|
| 7741202861 |
x = \gamma^2 x - \gamma^2 v t + \gamma v t'
|
|
|
|
|
| 7337056406 |
\gamma^2 x
|
|
|
|
|
| 4139999399 |
x - \gamma^2 x = - \gamma^2 v t + \gamma v t'
|
|
|
|
|
| 3495403335 |
x
|
|
|
|
|
| 9031609275 |
x (1 - \gamma^2 ) = - \gamma^2 v t + \gamma v t'
|
|
|
|
|
| 8014566709 |
\gamma^2 v t
|
|
|
|
|
| 9409776983 |
x (1 - \gamma^2 ) + \gamma^2 v t = \gamma v t'
|
|
|
|
|
| 2226340358 |
\gamma v
|
|
|
|
|
| 6417705759 |
\frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = \gamma v t'
|
|
|
|
|
| 8730201316 |
\frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t = t'
|
|
|
first term was multiplied by \gamma/\gamma
|
|
| 1974334644 |
\frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = t'
|
|
|
|
|
| 5148266645 |
t' = \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t
|
|
|
|
|
| 4287102261 |
x^2 + y^2 + z^2 = c^2 t^2
|
|
|
describes a spherical wavefront
|
|
| 1201689765 |
x'^2 + y'^2 + z'^2 = c^2 t'^2
|
|
|
describes a spherical wavefront for an observer in a moving frame of reference
|
|
| 7057864873 |
y' = y
|
|
|
frame of reference is moving only along x direction
|
|
| 8515803375 |
z' = z
|
|
|
frame of reference is moving only along x direction
|
|
| 6153463771 |
3
|
|
|
|
|
| 4746576736 |
asdf
|
|
|
|
|
| 1027270623 |
minga
|
|
|
|
|
| 6101803533 |
ming
|
|
|
|
|
| 9231495607 |
a = b
|
|
|
|
|
| 8664264194 |
b + 2
|
|
|
|
|
| 9805063945 |
\gamma^2 (x - v t)^2 + y^2 + z^2 = c^2 \gamma^2 \left( t + \frac{ 1 - \gamma^2 }{ \gamma^2 } \frac{x}{v} \right)^2
|
|
|
|
|
| 4057686137 |
C \to C_{\rm{Earth\ orbit}}
|
|
|
|
|
| 2346150725 |
r \to r_{\rm{Earth\ orbit}}
|
|
|
|
|
| 9753878784 |
v \to v_{\rm{Earth\ orbit}}
|
|
|
|
|
| 8135396036 |
t \to t_{\rm{Earth\ orbit}}
|
|
|
|
|
| 2773628333 |
\theta_1 \to \theta_{Brewster}
|
|
|
|
|
| 7154592211 |
\theta_2 \to \theta_{\rm refracted}
|
|
|
|
|
| 3749492596 |
E \to E_1
|
|
|
|
|
| 1258245373 |
E \to E_2
|
|
|
|
|
| 4147101187 |
KE \to KE_1
|
|
|
|
|
| 3809726424 |
PE \to PE_1
|
|
|
|
|
| 6383056612 |
KE \to KE_2
|
|
|
|
|
| 5075406409 |
PE \to PE_2
|
|
|
|
|
| 5398681503 |
v \to v_1
|
|
|
|
|
| 9305761407 |
v \to v_2
|
|
|
|
|
| 4218009993 |
x \to x_1
|
|
|
|
|
| 4188639044 |
x \to x_2
|
|
|
|
|
| 8173074178 |
x \to v_2
|
|
|
|
|
| 1025759423 |
y \to v_1
|
|
|
|
|
| 1935543849 |
\gamma^2 x^2 - \gamma^2 2 x v t + \gamma^2 v^2 t^2 + y^2 + z^2 = c^2 \gamma^2 \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x^2}{\gamma^2} + c^2 \gamma^2 2 t \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x}{\gamma} + c^2 \gamma^2 t^2
|
|
|
|
|
| 1586866563 |
\left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right) = t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)
|
|
|
|
|
| 3182633789 |
\gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1
|
|
|
|
|
| 1916173354 |
-\gamma^2 v^2 + c^2 \gamma^2 = c^2
|
|
|
|
|
| 2076171250 |
-\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} = 0
|
|
|
|
|
| 5284610349 |
\gamma^2
|
|
|
|
|
| 2417941373 |
- c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 - \gamma^2
|
|
|
|
|
| 5787469164 |
1 - \gamma^2
|
|
|
|
|
| 1639827492 |
- c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2} = 1
|
|
|
|
|
| 5669500954 |
v^2 \gamma^2
|
|
|
|
|
| 5763749235 |
-c^2 + c^2 \gamma^2 = v^2 \gamma^2
|
|
|
|
|
| 6408214498 |
c^2
|
|
|
|
|
| 2999795755 |
c^2 \gamma^2 = v^2 \gamma^2 + c^2
|
|
|
|
|
| 3412946408 |
v^2 \gamma^2
|
|
|
|
|
| 2542420160 |
c^2 \gamma^2 - v^2 \gamma^2 = c^2
|
|
|
|
|
| 7743841045 |
\gamma^2
|
|
|
|
|
| 7513513483 |
\gamma^2 (c^2 - v^2) = c^2
|
|
|
|
|
| 8571466509 |
c^2 - \gamma^2
|
|
|
|
|
| 7906112355 |
\gamma^2 = \frac{c^2}{c^2 - \gamma^2}
|
|
|
|
|
| 3060139639 |
1/2
|
|
|
|
|
| 1528310784 |
\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
|
|
|
|
|
| 8360117126 |
\gamma = \frac{-1}{\sqrt{1-\frac{v^2}{c^2}}}
|
|
|
not a physically valid result in this context
|
|
| 3366703541 |
a = \frac{v - v_0}{t}
|
|
|
acceleration is the average change in speed over a duration
|
|
| 7083390553 |
t
|
|
|
|
|
| 4748157455 |
a t = v - v_0
|
|
|
|
|
| 6417359412 |
v_0
|
|
|
|
|
| 4798787814 |
a t + v_0 = v
|
|
|
|
|
| 3462972452 |
v = v_0 + a t
|
|
|
|
|
| 3411994811 |
v_{\rm ave} = \frac{d}{t}
|
|
|
|
|
| 6175547907 |
v_{\rm ave} = \frac{v + v_0}{2}
|
|
|
|
|
| 9897284307 |
\frac{d}{t} = \frac{v + v_0}{2}
|
|
|
|
|
| 8865085668 |
t
|
|
|
|
|
| 8706092970 |
d = \left(\frac{v + v_0}{2}\right)t
|
|
|
|
|
| 7011114072 |
d = \frac{(v_0 + a t) + v_0}{2} t
|
|
|
|
|
| 1265150401 |
d = \frac{2 v_0 + a t}{2} t
|
|
|
|
|
| 9658195023 |
d = v_0 t + \frac{1}{2} a t^2
|
|
|
|
|
| 5799753649 |
2
|
|
|
|
|
| 7215099603 |
v^2 = v_0^2 + 2 a t v_0 + a^2 t^2
|
|
|
|
|
| 5144263777 |
v^2 = v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right)
|
|
|
|
|
| 7939765107 |
v^2 = v_0^2 + 2 a d
|
|
|
|
|
| 6729698807 |
v_0
|
|
|
|
|
| 9759901995 |
v - v_0 = a t
|
|
|
|
|
| 2242144313 |
a
|
|
|
|
|
| 1967582749 |
t = \frac{v - v_0}{a}
|
|
|
|
|
| 5733721198 |
d = \frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right)
|
|
|
|
|
| 5611024898 |
d = \frac{1}{2 a} (v^2 - v_0^2)
|
|
|
|
|
| 5542390646 |
2 a
|
|
|
|
|
| 8269198922 |
2 a d = v^2 - v_0^2
|
|
|
|
|
| 9070454719 |
v_0^2
|
|
|
|
|
| 4948763856 |
2 a d + v_0^2 = v^2
|
|
|
|
|
| 9645178657 |
a t
|
|
|
|
|
| 6457044853 |
v - a t = v_0
|
|
|
|
|
| 1259826355 |
d = (v - a t) t + \frac{1}{2} a t^2
|
|
|
|
|
| 4580545876 |
d = v t - a t^2 + \frac{1}{2} a t^2
|
|
|
|
|
| 6421241247 |
d = v t - \frac{1}{2} a t^2
|
|
|
|
|
| 4182362050 |
Z = |Z| \exp( i \theta )
|
|
|
Z \in \mathbb{C}
|
|
| 1928085940 |
Z^* = |Z| \exp( -i \theta )
|
|
|
|
|
| 9191880568 |
Z Z^* = |Z| |Z| \exp( -i \theta ) \exp( i \theta )
|
|
|
|
|
| 3350830826 |
Z Z^* = |Z|^2
|
|
|
|
|
| 8396997949 |
I = | A + B |^2
|
|
|
intensity of two waves traveling opposite directions on same path
|
|
| 4437214608 |
Z
|
|
|
|
|
| 5623794884 |
A + B
|
|
|
|
|
| 2236639474 |
(A + B)(A + B)^* = |A + B|^2
|
|
|
|
|
| 1020854560 |
I = (A + B)(A + B)^*
|
|
|
|
|
| 6306552185 |
I = (A + B)(A^* + B^*)
|
|
|
|
|
| 8065128065 |
I = A A^* + B B^* + A B^* + B A^*
|
|
|
|
|
| 9761485403 |
Z
|
|
|
|
|
| 8710504862 |
A
|
|
|
|
|
| 4075539836 |
A A^* = |A|^2
|
|
|
|
|
| 6529120965 |
B
|
|
|
|
|
| 1511199318 |
Z
|
|
|
|
|
| 7107090465 |
B B^* = |B|^2
|
|
|
|
|
| 5125940051 |
I = |A|^2 + B B^* + A B^* + B A^*
|
|
|
|
|
| 1525861537 |
I = |A|^2 + |B|^2 + A B^* + B A^*
|
|
|
|
|
| 1742775076 |
Z \to B
|
|
|
|
|
| 7607271250 |
\theta \to \phi
|
|
|
|
|
| 4192519596 |
B = |B| \exp(i \phi)
|
|
|
|
|
| 2064205392 |
A
|
|
|
|
|
| 1894894315 |
Z
|
|
|
|
|
| 1357848476 |
A = |A| \exp(i \theta)
|
|
|
|
|
| 6555185548 |
A^* = |A| \exp(-i \theta)
|
|
|
|
|
| 4504256452 |
B^* = |B| \exp(-i \phi)
|
|
|
|
|
| 7621705408 |
I = |A|^2 + |B|^2 + |A| |B| \exp(-i \theta) \exp(i \phi) + |A| |B| \exp(i \theta) \exp(-i \phi)
|
|
|
|
|
| 3085575328 |
I = |A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi))
|
|
|
|
|
| 4935235303 |
x
|
|
|
|
|
| 2293352649 |
\theta - \phi
|
|
|
|
|
| 3660957533 |
\cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
|
|
|
|
|
| 3967985562 |
2
|
|
|
|
|
| 2700934933 |
2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
|
|
|
|
|
| 8497631728 |
I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )
|
|
|
|
|
| 6774684564 |
\theta = \phi
|
|
|
for coherent waves
|
|
| 2099546947 |
I = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )
|
|
|
|
|
| 2719691582 |
|A| = |B|
|
|
|
in a loop
|
|
| 3257276368 |
I = |A|^2 + |A|^2 + |A| |A| 2
|
|
|
|
|
| 8283354808 |
I_{\rm coherent} = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )
|
|
|
|
|
| 8046208134 |
I_{\rm coherent} = |A|^2 + |A|^2 + |A| |A| 2
|
|
|
|
|
| 1172039918 |
I_{\rm coherent} = 4 |A|^2
|
|
|
|
|
| 8602221482 |
\langle \cos(\theta - \phi) \rangle = 0
|
|
|
incoherent light source
|
|
| 6240206408 |
I_{\rm incoherent} = |A|^2 + |B|^2
|
|
|
|
|
| 6529793063 |
I_{\rm incoherent} = |A|^2 + |A|^2
|
|
|
|
|
| 3060393541 |
I_{\rm incoherent} = 2|A|^2
|
|
|
|
|
| 6556875579 |
\frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2
|
|
|
|
|
| 8750500372 |
f
|
|
|
|
|
| 7543102525 |
g
|
|
|
|
|
| 8267133829 |
a = b
|
|
|
|
|
| 7968822469 |
c = d
|
|
|
|
|
| 3169580383 |
\vec{a} = \frac{d\vec{v}}{dt}
|
|
|
acceleration is the change in speed over a duration
|
|
| 8602512487 |
\vec{a} = a_x \hat{x} + a_y \hat{y}
|
|
|
decompose acceleration into two components
|
|
| 4158986868 |
a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt}
|
|
|
|
|
| 5349866551 |
\vec{v} = v_x \hat{x} + v_y \hat{y}
|
|
|
|
|
| 7729413831 |
a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right)
|
|
|
|
|
| 1819663717 |
a_x = \frac{d}{dt} v_x
|
|
|
|
|
| 8228733125 |
a_y = \frac{d}{dt} v_y
|
|
|
|
|
| 9707028061 |
a_x = 0
|
|
|
|
|
| 2741489181 |
a_y = -g
|
|
|
|
|
| 3270039798 |
2
|
|
|
|
|
| 8880467139 |
2
|
|
|
|
|
| 8750379055 |
0 = \frac{d}{dt} v_x
|
|
|
|
|
| 1977955751 |
-g = \frac{d}{dt} v_y
|
|
|
|
|
| 6672141531 |
dt
|
|
|
|
|
| 1702349646 |
-g dt = d v_y
|
|
|
|
|
| 8584698994 |
-g \int dt = \int d v_y
|
|
|
|
|
| 9973952056 |
-g t = v_y - v_{0, y}
|
|
|
|
|
| 4167526462 |
v_{0, y}
|
|
|
|
|
| 6572039835 |
-g t + v_{0, y} = v_y
|
|
|
|
|
| 7252338326 |
v_y = \frac{dy}{dt}
|
|
|
|
|
| 6204539227 |
-g t + v_{0, y} = \frac{dy}{dt}
|
|
|
|
|
| 1614343171 |
dt
|
|
|
|
|
| 8145337879 |
-g t dt + v_{0, y} dt = dy
|
|
|
|
|
| 8808860551 |
-g \int t dt + v_{0, y} \int dt = \int dy
|
|
|
|
|
| 2858549874 |
- \frac{1}{2} g t^2 + v_{0, y} t = y - y_0
|
|
|
|
|
| 6098638221 |
y_0
|
|
|
|
|
| 2461349007 |
- \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y
|
|
|
|
|
| 8717193282 |
dt
|
|
|
|
|
| 1166310428 |
0 dt = d v_x
|
|
|
|
|
| 2366691988 |
\int 0 dt = \int d v_x
|
|
|
|
|
| 1676472948 |
0 = v_x - v_{0, x}
|
|
|
|
|
| 1439089569 |
v_{0, x}
|
|
|
|
|
| 6134836751 |
v_{0, x} = v_x
|
|
|
|
|
| 8460820419 |
v_x = \frac{dx}{dt}
|
|
|
|
|
| 7455581657 |
v_{0, x} = \frac{dx}{dt}
|
|
|
|
|
| 8607458157 |
dt
|
|
|
|
|
| 1963253044 |
v_{0, x} dt = dx
|
|
|
|
|
| 3676159007 |
v_{0, x} \int dt = \int dx
|
|
|
|
|
| 9882526611 |
v_{0, x} t = x - x_0
|
|
|
|
|
| 3182907803 |
x_0
|
|
|
|
|
| 8486706976 |
v_{0, x} t + x_0 = x
|
|
|
|
|
| 1306360899 |
x = v_{0, x} t + x_0
|
|
|
|
|
| 7049769409 |
2
|
|
|
|
|
| 9341391925 |
\vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y}
|
|
|
|
|
| 6410818363 |
\theta
|
|
|
|
|
| 7391837535 |
\cos(\theta) = \frac{v_{0, x}}{v_0}
|
|
|
|
|
| 7376526845 |
\sin(\theta) = \frac{v_{0, y}}{v_0}
|
|
|
|
|
| 5868731041 |
v_0
|
|
|
|
|
| 6083821265 |
v_0 \cos(\theta) = v_{0, x}
|
|
|
|
|
| 5438722682 |
x = v_0 t \cos(\theta) + x_0
|
|
|
|
|
| 5620558729 |
v_0
|
|
|
|
|
| 8949329361 |
v_0 \sin(\theta) = v_{0, y}
|
|
|
|
|
| 1405465835 |
y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0
|
|
|
|
|
| 9862900242 |
y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0
|
|
|
|
|
| 6050070428 |
v_{0, x}
|
|
|
|
|
| 3274926090 |
t = \frac{x - x_0}{v_{0, x}}
|
|
|
|
|
| 7354529102 |
y = - \frac{1}{2} g \left( \frac{x - x_0}{v_{0, x}} \right)^2 + v_{0, y} \frac{x - x_0}{v_{0, x}} + y_0
|
|
|
|
|
| 8406170337 |
y \to y_f
|
|
|
|
|
| 2403773761 |
t \to t_f
|
|
|
|
|
| 5379546684 |
y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
|
|
|
|
|
| 5373931751 |
t = t_f
|
|
|
|
|
| 9112191201 |
y_f = 0
|
|
|
|
|
| 8198310977 |
0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
|
|
|
|
|
| 1650441634 |
y_0 = 0
|
|
|
define coordinate system such that initial height is at origin
|
|
| 1087417579 |
0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)
|
|
|
|
|
| 4829590294 |
t_f
|
|
|
|
|
| 2086924031 |
0 = - \frac{1}{2} g t_f + v_0 \sin(\theta)
|
|
|
|
|
| 6974054946 |
\frac{1}{2} g t_f
|
|
|
|
|
| 1191796961 |
\frac{1}{2} g t_f = v_0 \sin(\theta)
|
|
|
|
|
| 2510804451 |
2/g
|
|
|
|
|
| 4778077984 |
t_f = \frac{2 v_0 \sin(\theta)}{g}
|
|
|
|
|
| 3273630811 |
x \to x_f
|
|
|
|
|
| 6732786762 |
t \to t_f
|
|
|
|
|
| 3485125659 |
x_f = v_0 t_f \cos(\theta) + x_0
|
|
|
|
|
| 4370074654 |
t = t_f
|
|
|
|
|
| 2378095808 |
x_f = x_0 + d
|
|
|
|
|
| 4268085801 |
x_0 + d = v_0 t_f \cos(\theta) + x_0
|
|
|
|
|
| 8072682558 |
x_0
|
|
|
|
|
| 7233558441 |
d = v_0 t_f \cos(\theta)
|
|
|
|
|
| 2297105551 |
d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)
|
|
|
|
|
| 7587034465 |
\theta
|
|
|
|
|
| 7214442790 |
x
|
|
|
|
|
| 2519058903 |
\sin(2 \theta) = 2 \sin(\theta) \cos(\theta)
|
|
|
|
|
| 8922441655 |
d = \frac{v_0^2}{g} \sin(2 \theta)
|
|
|
|
|
| 5667870149 |
\theta
|
|
|
|
|
| 1541916015 |
\theta = \frac{\pi}{4}
|
|
|
|
|
| 3607070319 |
d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)
|
|
|
|
|
| 5353282496 |
d = \frac{v_0^2}{g}
|
|
|
|
|
| 0000040490 |
a^2
|
|
|
|
|
| 0000999900 |
b/(2a)
|
|
|
|
|
| 0001030901 |
\cos(x)
|
|
|
|
|
| 0001111111 |
(\sin(x))^2
|
|
|
|
|
| 0001209482 |
2 \pi
|
|
|
|
|
| 0001304952 |
\hbar
|
|
|
|
|
| 0001334112 |
W
|
|
|
|
|
| 0001921933 |
2 i
|
|
|
|
|
| 0002239424 |
2
|
|
|
|
|
| 0002338514 |
\vec{p}_{2}
|
|
|
|
|
| 0002342425 |
m/m
|
|
|
|
|
| 0002367209 |
1/2
|
|
|
|
|
| 0002393922 |
x
|
|
|
|
|
| 0002424922 |
a
|
|
|
|
|
| 0002436656 |
i \hbar
|
|
|
|
|
| 0002449291 |
b/(2a)
|
|
|
|
|
| 0002838490 |
b/(2a)
|
|
|
|
|
| 0002919191 |
\sin(-x)
|
|
|
|
|
| 0002929944 |
1/2
|
|
|
|
|
| 0002940021 |
2 \pi
|
|
|
|
|
| 0003232242 |
t
|
|
|
|
|
| 0003413423 |
\cos(-x)
|
|
|
|
|
| 0003747849 |
-1
|
|
|
|
|
| 0003838111 |
2
|
|
|
|
|
| 0003919391 |
x
|
|
|
|
|
| 0003949052 |
-x
|
|
|
|
|
| 0003949921 |
\hbar
|
|
|
|
|
| 0003954314 |
dx
|
|
|
|
|
| 0003981813 |
-\sin(x)
|
|
|
|
|
| 0004089571 |
2x
|
|
|
|
|
| 0004264724 |
y
|
|
|
|
|
| 0004307451 |
(b/(2a))^2
|
|
|
|
|
| 0004582412 |
x
|
|
|
|
|
| 0004829194 |
2
|
|
|
|
|
| 0004831494 |
a
|
|
|
|
|
| 0004849392 |
x
|
|
|
|
|
| 0004858592 |
h
|
|
|
|
|
| 0004934845 |
x
|
|
|
|
|
| 0004948585 |
a
|
|
|
|
|
| 0005395034 |
a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
|
|
|
|
|
| 0005626421 |
t
|
|
|
|
|
| 0005749291 |
f
|
|
|
|
|
| 0005938585 |
\frac{-\hbar^2}{2m}
|
|
|
|
|
| 0006466214 |
(\sin(x))^2
|
|
|
|
|
| 0006544644 |
t
|
|
|
|
|
| 0006563727 |
x
|
|
|
|
|
| 0006644853 |
c/a
|
|
|
|
|
| 0006656532 |
e
|
|
|
|
|
| 0007471778 |
2(\sin(x))^2
|
|
|
|
|
| 0007563791 |
i
|
|
|
|
|
| 0007636749 |
x
|
|
|
|
|
| 0007894942 |
(\sin(x))^2
|
|
|
|
|
| 0008837284 |
T
|
|
|
|
|
| 0008842811 |
\cos(2x)
|
|
|
|
|
| 0009458842 |
\psi(x)
|
|
|
|
|
| 0009484724 |
\frac{n \pi}{W}x
|
|
|
|
|
| 0009485857 |
a^2\frac{2}{W}
|
|
|
|
|
| 0009485858 |
\frac{2n\pi}{W}
|
|
|
|
|
| 0009492929 |
v du
|
|
|
|
|
| 0009587738 |
\psi
|
|
|
|
|
| 0009594995 |
1/2
|
|
|
|
|
| 0009877781 |
y
|
|
|
|
|
| 0203024440 |
1 = \int_0^W a \sin(\frac{n \pi}{W} x) \psi(x)^* dx
|
|
|
|
|
| 0404050504 |
\lambda = \frac{v}{f}
|
|
|
|
|
| 0439492440 |
\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)\right|_0^W
|
|
|
|
|
| 0934990943 |
k = \frac{2 \pi}{v T}
|
|
|
|
|
| 0948572140 |
\int \cos(a x) dx = \frac{1}{a}\sin(a x)
|
|
|
|
|
| 1010393913 |
\langle \psi| \hat{A}^+ |\psi \rangle = \langle a \rangle^*
|
|
|
|
|
| 1010393944 |
x = \langle\psi_{\alpha}| a_{\beta} |\psi_{\beta} \rangle
|
|
|
|
|
| 1010923823 |
k W = n \pi
|
|
|
|
|
| 1020010291 |
0 = a \sin(k W)
|
|
|
|
|
| 1020394900 |
p = h/\lambda
|
|
|
|
|
| 1020394902 |
E = h f
|
|
|
|
|
| 1029039903 |
p = m v
|
|
|
|
|
| 1029039904 |
p^2 = m^2 v^2
|
|
|
|
|
| 1158485859 |
\frac{-\hbar^2}{2m} \nabla^2 = {\cal H}
|
|
|
|
|
| 1202310110 |
\frac{1}{a^2} = \int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
|
|
|
|
|
| 1202312210 |
\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
|
|
|
|
|
| 1203938249 |
a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle = a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
|
|
|
|
|
| 1248277773 |
\cos(2x) = 1-2(\sin(x))^2
|
|
|
|
|
| 1293913110 |
0 = b
|
|
|
|
|
| 1293923844 |
\lambda = v T
|
|
|
|
|
| 1314464131 |
\vec{\nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
|
|
|
|
|
| 1314864131 |
\vec{\nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E}
|
|
|
|
|
| 1395858355 |
x = \langle \psi_{\alpha}| a_{\alpha} |\psi_{\beta}\rangle
|
|
|
|
|
| 1492811142 |
f = x - d
|
|
|
|
|
| 1492842000 |
\nabla \vec{x} = f(y)
|
|
|
|
|
| 1636453295 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = - \nabla^2 \vec{E}
|
|
|
|
|
| 1638282134 |
\vec{p}_{before} = \vec{p}_{after}
|
|
|
|
|
| 1648958381 |
\nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{\nabla}\psi(\vec{r},t)\right)
|
|
|
|
|
| 1857710291 |
0 = a \sin(n \pi)
|
|
|
|
|
| 1858578388 |
\nabla^2 E(\vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 1858772113 |
k = \frac{n \pi}{W}
|
|
|
|
|
| 1934748140 |
\int |\psi(x)|^2 dx = 1
|
|
|
|
|
| 2029293929 |
\nabla^2 E(\vec{r})\exp(i \omega t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 2103023049 |
\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right)
|
|
|
|
|
| 2113211456 |
f = 1/T
|
|
|
|
|
| 2123139121 |
-\exp(-i x) = -\cos(x)+i \sin(x)
|
|
|
|
|
| 2131616531 |
T f = 1
|
|
|
|
|
| 2258485859 |
{\cal H} \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
|
|
|
|
|
| 2394240499 |
x = a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle
|
|
|
|
|
| 2394853829 |
\exp(-i x) = \cos(-x)+i \sin(-x)
|
|
|
|
|
| 2394935831 |
( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
|
|
|
|
|
| 2394935835 |
\left(\langle\psi| \hat{A} |\psi\rangle \right)^+ = \left(\langle a \rangle\right)^+
|
|
|
|
|
| 2395958385 |
\nabla^2 \psi\left(\vec{r},t)\right) = \frac{-p^2}{\hbar} \psi(\vec{r},t)
|
|
|
|
|
| 2404934990 |
\langle x^2\rangle -2\langle x \rangle\langle x \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 2494533900 |
\langle x^2\rangle -\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 2569154141 |
\vec{\nabla} \times \frac{\partial}{\partial t}\vec{H} = \epsilon_0 \frac{\partial^2 }{\partial t^2}\vec{E}
|
|
|
|
|
| 2648958382 |
\nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p}\psi(\vec{r},t)\right)
|
|
|
|
|
| 2848934890 |
\langle a \rangle^* = \langle a \rangle
|
|
|
|
|
| 2944838499 |
\psi(x) = a \sin(\frac{n \pi}{W} x)
|
|
|
|
|
| 3121234211 |
\frac{k}{2\pi} = \lambda
|
|
|
|
|
| 3121234212 |
p = \frac{h k}{2\pi}
|
|
|
|
|
| 3121513111 |
k = \frac{2 \pi}{\lambda}
|
|
|
|
|
| 3131111133 |
T = 1 / f
|
|
|
|
|
| 3131211131 |
\omega = 2 \pi f
|
|
|
|
|
| 3132131132 |
\omega = \frac{2\pi}{T}
|
|
|
|
|
| 3147472131 |
\frac{\omega}{2 \pi} = f
|
|
|
|
|
| 3285732911 |
(\cos(x))^2 = 1-(\sin(x))^2
|
|
|
|
|
| 3485475729 |
\nabla^2 E(\vec{r}) = - \frac{\omega^2}{c^2} E(\vec{r})
|
|
|
|
|
| 3585845894 |
\langle \left(x-\langle x \rangle\right)^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 3829492824 |
\frac{1}{2}\left(\exp(i x)+\exp(-i x)\right) = \cos(x)
|
|
|
|
|
| 3924948349 |
a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
|
|
|
|
|
| 3942849294 |
\exp(i x)-\exp(-i x) = 2 i \sin(x)
|
|
|
|
|
| 3943939590 |
x = a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle
|
|
|
|
|
| 3947269979 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
|
|
|
|
|
| 3948571256 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi(\vec{r},t)
|
|
|
|
|
| 3948572230 |
\vec{\nabla}\psi(\vec{r},t) = \psi_0 \vec{\nabla}\exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
|
|
|
|
|
| 3948574224 |
\psi(\vec{r},t) = \psi_0 \exp\left(i\left(\vec{k}\cdot\vec{r} - \omega t \right) \right)
|
|
|
|
|
| 3948574226 |
\psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right)
|
|
|
|
|
| 3948574228 |
\psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
|
|
|
|
|
| 3948574230 |
\psi(\vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
|
|
|
|
|
| 3948574233 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
|
|
|
|
|
| 3948574235 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
|
|
|
|
|
| 3951205425 |
\vec{p}_{after} = \vec{p}_{1}
|
|
|
|
|
| 4147472132 |
E = \frac{h \omega}{2 \pi}
|
|
|
|
|
| 4298359835 |
E = \frac{1}{2}m v^2
|
|
|
|
|
| 4298359845 |
E = \frac{1}{2m}m^2 v^2
|
|
|
|
|
| 4298359851 |
E = \frac{p^2}{2m}
|
|
|
|
|
| 4341171256 |
i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{p^2}{2 m} \psi(\vec{r},t)
|
|
|
|
|
| 4348571256 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi(\vec{r},t)
|
|
|
|
|
| 4394958389 |
\vec{\nabla}\cdot \left(\vec{\nabla}\psi(\vec{r},t)\right) = \frac{i}{\hbar} \vec{\nabla}\cdot\left( \vec{p} \psi(\vec{r},t)\right)
|
|
|
|
|
| 4585828572 |
\epsilon_0 \mu_0 = \frac{1}{c^2}
|
|
|
|
|
| 4585932229 |
\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
|
|
|
|
|
| 4598294821 |
\exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
|
|
|
|
|
| 4638429483 |
\exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))
|
|
|
|
|
| 4742644828 |
\exp(i x)+\exp(-i x) = 2 \cos(x)
|
|
|
|
|
| 4827492911 |
\cos(2x)+(\sin(x))^2 = 1-(\sin(x))^2
|
|
|
|
|
| 4838429483 |
\exp(2 i x) = \cos(2 x)+i \sin(2 x)
|
|
|
|
|
| 4843995999 |
\frac{1}{2 i}\left(\exp(i x)-\exp(-i x)\right) = \sin(x)
|
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|
|
|
| 4857472413 |
1 = \int \psi(x)\psi(x)^* dx
|
|
|
|
|
| 4857475848 |
\frac{1}{a^2} = \frac{W}{2}
|
|
|
|
|
| 4858429483 |
\exp(i x)\exp(i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))
|
|
|
|
|
| 4923339482 |
i x = \log(y)
|
|
|
|
|
| 4928239482 |
\log(y) = i x
|
|
|
|
|
| 4928923942 |
a = b
|
|
|
|
|
| 4929218492 |
a+b = c
|
|
|
|
|
| 4929482992 |
b = 2
|
|
|
|
|
| 4938429482 |
\exp(-i x) = \cos(x)+i \sin(-x)
|
|
|
|
|
| 4938429483 |
\exp(i x) = \cos(x)+i \sin(x)
|
|
|
|
|
| 4938429484 |
\exp(-i x) = \cos(x)-i \sin(x)
|
|
|
|
|
| 4943571230 |
\vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
|
|
|
|
|
| 4948934890 |
\langle \psi| \hat{A} |\psi\rangle = \langle a \rangle^*
|
|
|
|
|
| 4949359835 |
\langle x^2\rangle -2\langle x^2 \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 4954839242 |
\cos(2x)+i\sin(2x) = (\cos(x)+i \sin(x))(\cos(x)+i \sin(x))
|
|
|
|
|
| 4978429483 |
\exp(-i x) = \cos(-x)+i \sin(-x)
|
|
|
|
|
| 4984892984 |
a+2 = c
|
|
|
|
|
| 4985825552 |
\nabla^2 E(\vec{r})\exp(i \omega t) = i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 5530148480 |
\vec{p}_{1}-\vec{p}_{2} = \vec{p}_{electron}
|
|
|
|
|
| 5727578862 |
\frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)
|
|
|
|
|
| 5832984291 |
(\sin(x))^2 + (\cos(x))^2 = 1
|
|
|
|
|
| 5857434758 |
\int a dx = a x
|
|
|
|
|
| 5868688585 |
\frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = \frac{p^2}{2m} \psi(\vec{r},t)
|
|
|
|
|
| 5900595848 |
k = \frac{\omega}{v}
|
|
|
|
|
| 5928285821 |
x^2 + 2x(b/(2a)) + (b/(2a))^2 = (x+(b/(2a)))^2
|
|
|
|
|
| 5928292841 |
x^2 + (b/a)x + (b/(2a))^2 = -c/a + (b/(2a))^2
|
|
|
|
|
| 5938459282 |
x^2 + (b/a)x = -c/a
|
|
|
|
|
| 5958392859 |
x^2 + (b/a)x+(c/a) = 0
|
|
|
|
|
| 5959282914 |
x^2 + x(b/a) + (b/(2a))^2 = (x+(b/(2a)))^2
|
|
|
|
|
| 5982958248 |
x = -\sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
|
|
|
|
|
| 5982958249 |
x+(b/(2a)) = -\sqrt{(b/(2a))^2 - (c/a)}
|
|
|
|
|
| 5985371230 |
\vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi(\vec{r},t)
|
|
|
|
|
| 6742123016 |
\vec{p}_{electron}\cdot\vec{p}_{electron} = (\vec{p}_{1}\cdot\vec{p}_{1})+(\vec{p}_{2}\cdot\vec{p}_{2})-2(\vec{p}_{1}\cdot\vec{p}_{2})
|
|
|
|
|
| 7466829492 |
\vec{\nabla} \cdot \vec{E} = 0
|
|
|
|
|
| 7564894985 |
\int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)
|
|
|
|
|
| 7572664728 |
\cos(2x)+2(\sin(x))^2 = 1
|
|
|
|
|
| 7575738420 |
\left(\sin\left(\frac{n \pi}{W}x\right)\right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2}
|
|
|
|
|
| 7575859295 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859300 |
\epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{\nabla} \times \vec{E} )_k = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859302 |
\epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859304 |
\epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h}
|
|
|
|
|
| 7575859306 |
\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859308 |
\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right) = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859310 |
\hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859312 |
\vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7917051060 |
\vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2}
|
|
|
|
|
| 8139187332 |
\vec{p}_{1} = \vec{p}_{2}+\vec{p}_{electron}
|
|
|
|
|
| 8257621077 |
\vec{p}_{before} = \vec{p}_{1}
|
|
|
|
|
| 8311458118 |
\vec{p}_{after} = \vec{p}_{2}+\vec{p}_{electron}
|
|
|
|
|
| 8399484849 |
\langle x^2 -2x\langle x \rangle+\langle x \rangle^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 8484544728 |
-a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x)
|
|
|
|
|
| 8485757728 |
a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(kx)
|
|
|
|
|
| 8485867742 |
\frac{2}{W} = a^2
|
|
|
|
|
| 8489593958 |
d(u v) = u dv + v du
|
|
|
|
|
| 8489593960 |
d(u v) - v du = u dv
|
|
|
|
|
| 8489593962 |
u dv = d(u v) - v du
|
|
|
|
|
| 8489593964 |
\int u dv = u v - \int v du
|
|
|
|
|
| 8494839423 |
\nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
|
|
|
|
|
| 8572657110 |
1 = \int |\psi(x)|^2 dx
|
|
|
|
|
| 8572852424 |
\vec{E} = E(\vec{r},t)
|
|
|
|
|
| 8575746378 |
\int \frac{1}{2} dx = \frac{1}{2} x
|
|
|
|
|
| 8575748999 |
\frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x)\right) = -k^2 \left(a \sin(kx) + b \cos(kx)\right)
|
|
|
|
|
| 8576785890 |
1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
|
|
|
|
|
| 8577275751 |
0 = a \sin(0) + b\cos(0)
|
|
|
|
|
| 8582885111 |
\psi(x) = a \sin(kx) + b \cos(kx)
|
|
|
|
|
| 8582954722 |
x^2 + 2xh + h^2 = (x+h)^2
|
|
|
|
|
| 8849289982 |
\psi(x)^* = a \sin(\frac{n \pi}{W} x)
|
|
|
|
|
| 8889444440 |
1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right)\right)^2 dx
|
|
|
|
|
| 9059289981 |
\psi(x) = a \sin(k x)
|
|
|
|
|
| 9285928292 |
ax^2 + bx + c = 0
|
|
|
|
|
| 9291999979 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0\vec{\nabla} \times \frac{\partial \vec{H}}{\partial t}
|
|
|
|
|
| 9294858532 |
\hat{A}^+ = \hat{A}
|
|
|
|
|
| 9385938295 |
(x+(b/(2a)))^2 = -(c/a) + (b/(2a))^2
|
|
|
|
|
| 9393939991 |
\psi(x) = -\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
|
|
|
|
|
| 9393939992 |
\psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
|
|
|
|
|
| 9394939493 |
\nabla^2 E(\vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r},t)
|
|
|
|
|
| 9429829482 |
\frac{d}{dx} y = -\sin(x) + i\cos(x)
|
|
|
|
|
| 9482113948 |
\frac{dy}{y} = i dx
|
|
|
|
|
| 9482438243 |
(\cos(x))^2 = \cos(2x)+(\sin(x))^2
|
|
|
|
|
| 9482923849 |
\exp(i x) = y
|
|
|
|
|
| 9482928242 |
\cos(2x) = (\cos(x))^2 - (\sin(x))^2
|
|
|
|
|
| 9482928243 |
\cos(2x)+(\sin(x))^2 = (\cos(x))^2
|
|
|
|
|
| 9482943948 |
\log(y) = i dx
|
|
|
|
|
| 9482948292 |
b = c
|
|
|
|
|
| 9482984922 |
\frac{d}{dx} y = (i\sin(x) + \cos(x)) i
|
|
|
|
|
| 9483928192 |
\cos(2x)+i\sin(2x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
|
|
|
|
|
| 9485384858 |
\nabla^2 E(\vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 9485747245 |
\sqrt{\frac{2}{W}} = a
|
|
|
|
|
| 9485747246 |
-\sqrt{\frac{2}{W}} = a
|
|
|
|
|
| 9492920340 |
y = \cos(x)+i \sin(x)
|
|
|
|
|
| 9494829190 |
k
|
|
|
|
|
| 9495857278 |
\psi(x=W) = 0
|
|
|
|
|
| 9499428242 |
E(\vec{r},t) = E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 9499959299 |
a + k = b + k
|
|
|
|
|
| 9582958293 |
x = \sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
|
|
|
|
|
| 9582958294 |
x+(b/(2a)) = \sqrt{(b/(2a))^2 - (c/a)}
|
|
|
|
|
| 9584821911 |
c + d = a
|
|
|
|
|
| 9585727710 |
\psi(x=0) = 0
|
|
|
|
|
| 9596004948 |
x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle
|
|
|
|
|
| 9848292229 |
dy = y i dx
|
|
|
|
|
| 9848294829 |
\frac{d}{dx} y = y i
|
|
|
|
|
| 9858028950 |
\frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
|
|
|
|
|
| 9889984281 |
2(\sin(x))^2 = 1-\cos(2x)
|
|
|
|
|
| 9919999981 |
\rho = 0
|
|
|
|
|
| 9958485859 |
\frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
|
|
|
|
|
| 9988949211 |
(\sin(x))^2 = \frac{1-\cos(2x)}{2}
|
|
|
|
|
| 9991999979 |
\vec{\nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t}
|
|
|
|
|
| 9999998870 |
\frac{\vec{p}}{\hbar} = \vec{k}
|
|
|
|
|
| 9999999870 |
\frac{p}{\hbar} = k
|
|
|
|
|
| 9999999950 |
\beta = 1/(k_{Boltzmann}T)
|
|
|
|
|
| 9999999951 |
\langle x | k \rangle = \frac{\exp(i k x)}{\sqrt{2\pi}}
|
|
|
|
|
| 9999999952 |
f(x) \delta(x-a) = f(a)
|
|
|
|
|
| 9999999953 |
\int_{-\infty}^{\infty} \delta(x) dx = 1
|
|
|
|
|
| 9999999954 |
c = 1/\sqrt{\epsilon_0 \mu_0}
|
|
|
|
|
| 9999999955 |
\vec{E} = -\vec{\nabla} \Psi
|
|
|
|
|
| 9999999956 |
\vec{F} = \frac{\partial}{\partial t}\vec{p}
|
|
|
|
|
| 9999999957 |
\vec{F} = -\vec{\nabla} V
|
|
|
|
|
| 9999999960 |
\hbar = h/(2 \pi)
|
|
|
|
|
| 9999999961 |
\frac{E}{\hbar} = \omega
|
|
|
|
|
| 9999999962 |
p = \hbar k
|
|
|
|
|
| 9999999963 |
\lambda = h/p
|
|
|
|
|
| 9999999964 |
\omega = c k
|
|
|
|
|
| 9999999965 |
E = \omega \hbar
|
|
|
|
|
| 9999999966 |
\vec{L} = \vec{r}\times\vec{p}
|
|
|
|
|
| 9999999967 |
\vec{S} = \frac{1}{\mu_0} \vec{E}\times \vec{B}
|
|
|
|
|
| 9999999968 |
x = \frac{-b-\sqrt{b^2-4ac}}{2a}
|
|
|
|
|
| 9999999969 |
x = \frac{-b+\sqrt{b^2-4ac}}{2a}
|
|
|
|
|
| 9999999970 |
\eta_1 \sin\theta_1 = \eta_2 \sin\theta_2
|
|
|
|
|
| 9999999971 |
{\cal H} = \frac{p^2}{2m}+V
|
|
|
|
|
| 9999999972 |
{\cal H} |\psi\rangle = E |\psi\rangle
|
|
|
|
|
| 9999999973 |
\left( \Delta A \right)^2 = \langle A^2 \rangle - \langle A \rangle^2
|
|
|
|
|
| 9999999974 |
\langle \psi| \hat{A} |\psi\rangle = \int_{-\infty}^{\infty} \psi^* A \psi dx
|
|
|
|
|
| 9999999975 |
\langle \psi| \hat{A} |\psi\rangle = \langle a \rangle
|
|
|
|
|
| 9999999976 |
\hat{p} = -i \hbar \frac{\partial }{\partial x}
|
|
|
|
|
| 9999999977 |
[\hat{x},\hat{p}] = i \hbar
|
|
|
|
|
| 9999999978 |
\vec{\nabla} \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial E}{\partial t}
|
|
|
|
|
| 9999999979 |
\vec{\nabla} \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}
|
|
|
|
|
| 9999999980 |
\vec{\nabla} \cdot \vec{B} = 0
|
|
|
|
|
| 9999999981 |
\vec{\nabla} \cdot \vec{E} = \rho/\epsilon_0
|
|
|
|
|
| 9999999982 |
V = I R + Q/C + L \frac{\partial I}{\partial t}
|
|
|
|
|
| 9999999983 |
C = V A/d
|
|
|
|
|
| 9999999984 |
Q = C V
|
|
|
|
|
| 9999999985 |
V = I R
|
|
|
|
|
| 9999999986 |
\left[\right]\left[\right] = \left[\right]
|
|
|
|
|
| 9999999987 |
1 atmosphere = 101.325 Pascal
|
|
|
|
|
| 9999999988 |
1 atmosphere = 14.7 pounds/(inch^2)
|
|
|
|
|
| 9999999989 |
mass_{electron} = 511000 electronVolts/(q^2)
|
|
|
|
|
| 9999999990 |
Tesla = 10000*Gauss
|
|
|
|
|
| 9999999991 |
Pascal = Newton / (meter^2)
|
|
|
|
|
| 9999999992 |
Tesla = Newton / (Ampere*meter)
|
|
|
|
|
| 9999999993 |
Farad = Coulumb / Volt
|
|
|
|
|
| 9999999995 |
Volt = Joule / Coulumb
|
|
|
|
|
| 9999999996 |
Coulomb = Ampere / second
|
|
|
|
|
| 9999999997 |
Watt = Joule / second
|
|
|
|
|
| 9999999998 |
Joule = Newton*meter
|
|
|
|
|
| 9999999999 |
Newton = kilogram*meter/(second^2)
|
|
|
|
|
| 4961662865 |
x
|
|
|
|
|
| 9110536742 |
2 x
|
|
|
|
|
| 8483686863 |
\sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x)\right)
|
|
|
|
|
| 3470587782 |
\sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
|
|
|
|
|
| 8642992037 |
2
|
|
|
|
|
| 9894826550 |
2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \left(\exp(i x)+\exp(-i x)\right)
|
|
|
|
|
| 4257214632 |
\frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)
|
|
|
|
|
| 8699789241 |
2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)
|
|
|
|
|
| 9180861128 |
2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)
|
|
|
|
|
| 2405307372 |
\sin(2 x) = 2 \sin(x) \cos(x)
|
|
|
|
|
| 3268645065 |
x
|
|
|
|
|
| 9350663581 |
\pi
|
|
|
|
|
| 8332931442 |
\exp(i \pi) = \cos(\pi)+i \sin(\pi)
|
|
|
|
|
| 6885625907 |
\exp(i \pi) = -1 + i 0
|
|
|
|
|
| 3331824625 |
\exp(i \pi) = -1
|
|
|
|
|
| 4901237716 |
1
|
|
|
|
|
| 2501591100 |
\exp(i \pi) + 1 = 0
|
|
|
|
|
| 5136652623 |
E = KE + PE
|
|
mechanical energy is the sum of the potential plus kinetic energies |
|
|
| 7915321076 |
E, KE, PE
|
|
|
|
|
| 3192374582 |
E_2, KE_2, PE_2
|
|
|
|
|
| 7875206161 |
E_2 = KE_2 + PE_2
|
|
|
|
|
| 4229092186 |
E, KE, PE
|
|
|
|
|
| 1490821948 |
E_1, KE_1, PE_1
|
|
|
|
|
| 4303372136 |
E_1 = KE_1 + PE_1
|
|
|
|
|
| 5514556106 |
E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1)
|
|
|
|
|
| 8357234146 |
KE = \frac{1}{2} m v^2
|
|
kinetic energy |
Kinetic_energy
|
|
| 3658066792 |
KE_2, v_2
|
|
|
|
|
| 8082557839 |
KE, v
|
|
|
|
|
| 7676652285 |
KE_2 = \frac{1}{2} m v_2^2
|
|
|
|
|
| 2790304597 |
KE_1, v_1
|
|
|
|
|
| 9880327189 |
KE, v
|
|
|
|
|
| 4928007622 |
KE_1 = \frac{1}{2} m v_1^2
|
|
|
|
|
| 5733146966 |
KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right)
|
|
|
|
|
| 6554292307 |
t
|
|
|
|
|
| 4270680309 |
\frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t}
|
|
|
|
|
| 5781981178 |
x^2 - y^2 = (x+y)(x-y)
|
|
difference of squares |
Difference_of_two_squares
|
|
| 5946759357 |
v_2, v_1
|
|
|
|
|
| 6675701064 |
x, y
|
|
|
|
|
| 4648451961 |
v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1)
|
|
|
|
|
| 9356924046 |
\frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t}
|
|
|
|
|
| 9397152918 |
v = \frac{v_1 + v_2}{2}
|
|
average velocity |
|
|
| 2857430695 |
a = \frac{v_2 - v_1}{t}
|
|
acceleration |
|
|
| 7735737409 |
\frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t}
|
|
|
|
|
| 4784793837 |
\frac{KE_2 - KE_1}{t} = m v a
|
|
|
|
|
| 5345738321 |
F = m a
|
|
Newton's second law of motion |
Newton%27s_laws_of_motion#Newton's_second_law
|
|
| 2186083170 |
\frac{KE_2 - KE_1}{t} = v F
|
|
|
|
|
| 6715248283 |
PE = -F x
|
|
potential energy |
Potential_energy
|
|
| 3852078149 |
PE_2, x_2
|
|
|
|
|
| 7388921188 |
PE, x
|
|
|
|
|
| 2431507955 |
PE_2 = -F x_2
|
|
|
|
|
| 3412641898 |
PE_1, x_1
|
|
|
|
|
| 9937169749 |
PE, x
|
|
|
|
|
| 4669290568 |
PE_1 = -F x_1
|
|
|
|
|
| 7734996511 |
PE_2 - PE_1 = -F ( x_2 - x_1 )
|
|
|
|
|
| 2016063530 |
t
|
|
|
|
|
| 7267155233 |
\frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right)
|
|
|
|
|
| 9337785146 |
v = \frac{x_2 - x_1}{t}
|
|
average velocity |
|
|
| 4872970974 |
\frac{PE_2 - PE_1}{t} = -F v
|
|
|
|
|
| 2081689540 |
t
|
|
|
|
|
| 2770069250 |
\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t}
|
|
|
|
|
| 3591237106 |
\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v
|
|
|
|
|
| 1772416655 |
\frac{E_2 - E_1}{t} = v F - F v
|
|
|
|
|
| 1809909100 |
\frac{E_2 - E_1}{t} = 0
|
|
|
|
|
| 5778176146 |
t
|
|
|
|
|
| 3806977900 |
E_2 - E_1 = 0
|
|
|
|
|
| 5960438249 |
E_1
|
|
|
|
|
| 8558338742 |
E_2 = E_1
|
|
|
|
|
| 8945218208 |
\theta_{Brewster} + \theta_{refracted} = 90^{\circ}
|
|
|
based on figure 34-27 on page 824 in 2001_HRW
|
|
| 9025853427 |
\theta_{Brewster}
|
|
|
|
|
| 1310571337 |
\theta_{refracted} = 90^{\circ} - \theta_{Brewster}
|
|
|
|
|
| 6450985774 |
n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 )
|
|
Law of Refraction |
eq 34-44 on page 819 in 2001_HRW
|
|
| 1779497048 |
\theta_{Brewster}, \theta_{refraction}
|
|
|
|
|
| 7322344455 |
\theta_1, \theta_2
|
|
|
|
|
| 2575937347 |
n_1 \sin( \theta_{Brewster} ) = n_2 \sin( \theta_{refraction} )
|
|
|
|
|
| 7696214507 |
n_1 \sin( \theta_{Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{Brewster} )
|
|
|
|
|
| 8588429722 |
\sin( 90^{\circ} - x ) = \cos( x )
|
|
|
|
|
| 7375348852 |
\theta_{Brewster}
|
|
|
|
|
| 1512581563 |
x
|
|
|
|
|
| 6831637424 |
\sin( 90^{\circ} - \theta_{Brewster} ) = \cos( \theta_{Brewster} )
|
|
|
|
|
| 3061811650 |
n_1 \sin( \theta_{Brewster} ) = n_2 \cos( \theta_{Brewster} )
|
|
|
|
|
| 7857757625 |
n_1
|
|
|
|
|
| 9756089533 |
\sin( \theta_{Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{Brewster} )
|
|
|
|
|
| 5632428182 |
\cos( \theta_{Brewster} )
|
|
|
|
|
| 2768857871 |
\frac{\sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )} = \frac{n_2}{n_1}
|
|
|
|
|
| 4968680693 |
\tan( x ) = \frac{ \sin( x )}{\cos( x )}
|
|
|
|
|
| 7321695558 |
\theta_{Brewster}
|
|
|
|
|
| 9906920183 |
x
|
|
|
|
|
| 4501377629 |
\tan( \theta_{Brewster} ) = \frac{ \sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )}
|
|
|
|
|
| 3417126140 |
\tan( \theta_{Brewster} ) = \frac{ n_2 }{ n_1 }
|
|
|
|
|
| 5453995431 |
\arctan{ x }
|
|
|
|
|
| 6023986360 |
x
|
|
|
|
|
| 8495187962 |
\theta_{Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }
|
|
|
|
|
| 9040079362 |
f
|
|
|
|
|
| 9565166889 |
T
|
|
|
|
|
| 5426308937 |
v = \frac{d}{t}
|
|
|
|
|
| 7476820482 |
C
|
|
|
|
|
| 1277713901 |
d
|
|
|
|
|
| 6946088325 |
v = \frac{C}{t}
|
|
|
|
|
| 6785303857 |
C = 2 \pi r
|
|
|
|
|
| 4816195267 |
C, r
|
|
|
|
|
| 2113447367 |
C_{\rm{Earth\ orbit}}, r_{\rm{Earth\ orbit}}
|
|
|
|
|
| 6348260313 |
C_{\rm{Earth\ orbit}} = 2 \pi r_{\rm{Earth\ orbit}}
|
|
|
|
|
| 3847777611 |
C, v, t
|
|
|
|
|
| 6406487188 |
C_{\rm{Earth\ orbit}}, v_{\rm{Earth\ orbit}}, t_{\rm{Earth\ orbit}}
|
|
|
|
|
| 3046191961 |
v_{\rm{Earth\ orbit}} = \frac{C_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
|
|
|
|
|
| 3080027960 |
v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
|
|
|
|
|
| 7175416299 |
t_{\rm{Earth\ orbit}} = 1 {\rm{year}}
|
|
|
|
|
| 3219318145 |
\frac{365 {\rm days}}{1 {\rm year}} \frac{24 {\rm hours}}{1 {\rm day}} \frac{60 {\rm minutes}}{1 {\rm hour}} \frac{60 {\rm seconds}}{1 {\rm minute}}
|
|
|
|
|
| 8721295221 |
t_{\rm{Earth\ orbit}} = 3.16 10^7 {\rm{seconds}}
|
|
|
|
|
| 4593428198 |
v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{3.16\ 10^7 {\rm seconds}}
|
|
|
|
|
| 3472836147 |
r_{\rm{Earth\ orbit}} = 1.496\ 10^8 {\rm km}
|
|
|
|
|
| 6998364753 |
v_{\rm{Earth\ orbit}} = \frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}}
|
|
|
|
|
| 4180845508 |
v_{\rm{Earth\ orbit}} = 29.8 \frac{{\rm km}}{{\rm sec}}
|
|
|
|
|
| 4662369843 |
x' = \gamma (x - v t)
|
|
|
|
|
| 2983053062 |
x = \gamma (x' + v t')
|
|
|
|
|
| 3426941928 |
x = \gamma ( \gamma (x - v t) + v t' )
|
|
|
|
|
| 2096918413 |
x = \gamma ( \gamma x - \gamma v t + v t' )
|
|
|
|
|
| 7741202861 |
x = \gamma^2 x - \gamma^2 v t + \gamma v t'
|
|
|
|
|
| 7337056406 |
\gamma^2 x
|
|
|
|
|
| 4139999399 |
x - \gamma^2 x = - \gamma^2 v t + \gamma v t'
|
|
|
|
|
| 3495403335 |
x
|
|
|
|
|
| 9031609275 |
x (1 - \gamma^2 ) = - \gamma^2 v t + \gamma v t'
|
|
|
|
|
| 8014566709 |
\gamma^2 v t
|
|
|
|
|
| 9409776983 |
x (1 - \gamma^2 ) + \gamma^2 v t = \gamma v t'
|
|
|
|
|
| 2226340358 |
\gamma v
|
|
|
|
|
| 6417705759 |
\frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = \gamma v t'
|
|
|
|
|
| 8730201316 |
\frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t = t'
|
|
|
first term was multiplied by \gamma/\gamma
|
|
| 1974334644 |
\frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = t'
|
|
|
|
|
| 5148266645 |
t' = \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t
|
|
|
|
|
| 4287102261 |
x^2 + y^2 + z^2 = c^2 t^2
|
|
|
describes a spherical wavefront
|
|
| 1201689765 |
x'^2 + y'^2 + z'^2 = c^2 t'^2
|
|
|
describes a spherical wavefront for an observer in a moving frame of reference
|
|
| 7057864873 |
y' = y
|
|
|
frame of reference is moving only along x direction
|
|
| 8515803375 |
z' = z
|
|
|
frame of reference is moving only along x direction
|
|
| 6153463771 |
3
|
|
|
|
|
| 4746576736 |
asdf
|
|
|
|
|
| 1027270623 |
minga
|
|
|
|
|
| 6101803533 |
ming
|
|
|
|
|
| 9231495607 |
a = b
|
|
|
|
|
| 8664264194 |
b + 2
|
|
|
|
|
| 9805063945 |
\gamma^2 (x - v t)^2 + y^2 + z^2 = c^2 \gamma^2 \left( t + \frac{ 1 - \gamma^2 }{ \gamma^2 } \frac{x}{v} \right)^2
|
|
|
|
|
| 4057686137 |
C \to C_{\rm{Earth\ orbit}}
|
|
|
|
|
| 2346150725 |
r \to r_{\rm{Earth\ orbit}}
|
|
|
|
|
| 9753878784 |
v \to v_{\rm{Earth\ orbit}}
|
|
|
|
|
| 8135396036 |
t \to t_{\rm{Earth\ orbit}}
|
|
|
|
|
| 2773628333 |
\theta_1 \to \theta_{Brewster}
|
|
|
|
|
| 7154592211 |
\theta_2 \to \theta_{\rm refracted}
|
|
|
|
|
| 3749492596 |
E \to E_1
|
|
|
|
|
| 1258245373 |
E \to E_2
|
|
|
|
|
| 4147101187 |
KE \to KE_1
|
|
|
|
|
| 3809726424 |
PE \to PE_1
|
|
|
|
|
| 6383056612 |
KE \to KE_2
|
|
|
|
|
| 5075406409 |
PE \to PE_2
|
|
|
|
|
| 5398681503 |
v \to v_1
|
|
|
|
|
| 9305761407 |
v \to v_2
|
|
|
|
|
| 4218009993 |
x \to x_1
|
|
|
|
|
| 4188639044 |
x \to x_2
|
|
|
|
|
| 8173074178 |
x \to v_2
|
|
|
|
|
| 1025759423 |
y \to v_1
|
|
|
|
|
| 1935543849 |
\gamma^2 x^2 - \gamma^2 2 x v t + \gamma^2 v^2 t^2 + y^2 + z^2 = c^2 \gamma^2 \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x^2}{\gamma^2} + c^2 \gamma^2 2 t \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x}{\gamma} + c^2 \gamma^2 t^2
|
|
|
|
|
| 1586866563 |
\left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right) = t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)
|
|
|
|
|
| 3182633789 |
\gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1
|
|
|
|
|
| 1916173354 |
-\gamma^2 v^2 + c^2 \gamma^2 = c^2
|
|
|
|
|
| 2076171250 |
-\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} = 0
|
|
|
|
|
| 5284610349 |
\gamma^2
|
|
|
|
|
| 2417941373 |
- c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 - \gamma^2
|
|
|
|
|
| 5787469164 |
1 - \gamma^2
|
|
|
|
|
| 1639827492 |
- c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2} = 1
|
|
|
|
|
| 5669500954 |
v^2 \gamma^2
|
|
|
|
|
| 5763749235 |
-c^2 + c^2 \gamma^2 = v^2 \gamma^2
|
|
|
|
|
| 6408214498 |
c^2
|
|
|
|
|
| 2999795755 |
c^2 \gamma^2 = v^2 \gamma^2 + c^2
|
|
|
|
|
| 3412946408 |
v^2 \gamma^2
|
|
|
|
|
| 2542420160 |
c^2 \gamma^2 - v^2 \gamma^2 = c^2
|
|
|
|
|
| 7743841045 |
\gamma^2
|
|
|
|
|
| 7513513483 |
\gamma^2 (c^2 - v^2) = c^2
|
|
|
|
|
| 8571466509 |
c^2 - \gamma^2
|
|
|
|
|
| 7906112355 |
\gamma^2 = \frac{c^2}{c^2 - \gamma^2}
|
|
|
|
|
| 3060139639 |
1/2
|
|
|
|
|
| 1528310784 |
\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
|
|
|
|
|
| 8360117126 |
\gamma = \frac{-1}{\sqrt{1-\frac{v^2}{c^2}}}
|
|
|
not a physically valid result in this context
|
|
| 3366703541 |
a = \frac{v - v_0}{t}
|
|
|
acceleration is the average change in speed over a duration
|
|
| 7083390553 |
t
|
|
|
|
|
| 4748157455 |
a t = v - v_0
|
|
|
|
|
| 6417359412 |
v_0
|
|
|
|
|
| 4798787814 |
a t + v_0 = v
|
|
|
|
|
| 3462972452 |
v = v_0 + a t
|
|
|
|
|
| 3411994811 |
v_{\rm ave} = \frac{d}{t}
|
|
|
|
|
| 6175547907 |
v_{\rm ave} = \frac{v + v_0}{2}
|
|
|
|
|
| 9897284307 |
\frac{d}{t} = \frac{v + v_0}{2}
|
|
|
|
|
| 8865085668 |
t
|
|
|
|
|
| 8706092970 |
d = \left(\frac{v + v_0}{2}\right)t
|
|
|
|
|
| 7011114072 |
d = \frac{(v_0 + a t) + v_0}{2} t
|
|
|
|
|
| 1265150401 |
d = \frac{2 v_0 + a t}{2} t
|
|
|
|
|
| 9658195023 |
d = v_0 t + \frac{1}{2} a t^2
|
|
|
|
|
| 5799753649 |
2
|
|
|
|
|
| 7215099603 |
v^2 = v_0^2 + 2 a t v_0 + a^2 t^2
|
|
|
|
|
| 5144263777 |
v^2 = v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right)
|
|
|
|
|
| 7939765107 |
v^2 = v_0^2 + 2 a d
|
|
|
|
|
| 6729698807 |
v_0
|
|
|
|
|
| 9759901995 |
v - v_0 = a t
|
|
|
|
|
| 2242144313 |
a
|
|
|
|
|
| 1967582749 |
t = \frac{v - v_0}{a}
|
|
|
|
|
| 5733721198 |
d = \frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right)
|
|
|
|
|
| 5611024898 |
d = \frac{1}{2 a} (v^2 - v_0^2)
|
|
|
|
|
| 5542390646 |
2 a
|
|
|
|
|
| 8269198922 |
2 a d = v^2 - v_0^2
|
|
|
|
|
| 9070454719 |
v_0^2
|
|
|
|
|
| 4948763856 |
2 a d + v_0^2 = v^2
|
|
|
|
|
| 9645178657 |
a t
|
|
|
|
|
| 6457044853 |
v - a t = v_0
|
|
|
|
|
| 1259826355 |
d = (v - a t) t + \frac{1}{2} a t^2
|
|
|
|
|
| 4580545876 |
d = v t - a t^2 + \frac{1}{2} a t^2
|
|
|
|
|
| 6421241247 |
d = v t - \frac{1}{2} a t^2
|
|
|
|
|
| 4182362050 |
Z = |Z| \exp( i \theta )
|
|
|
Z \in \mathbb{C}
|
|
| 1928085940 |
Z^* = |Z| \exp( -i \theta )
|
|
|
|
|
| 9191880568 |
Z Z^* = |Z| |Z| \exp( -i \theta ) \exp( i \theta )
|
|
|
|
|
| 3350830826 |
Z Z^* = |Z|^2
|
|
|
|
|
| 8396997949 |
I = | A + B |^2
|
|
|
intensity of two waves traveling opposite directions on same path
|
|
| 4437214608 |
Z
|
|
|
|
|
| 5623794884 |
A + B
|
|
|
|
|
| 2236639474 |
(A + B)(A + B)^* = |A + B|^2
|
|
|
|
|
| 1020854560 |
I = (A + B)(A + B)^*
|
|
|
|
|
| 6306552185 |
I = (A + B)(A^* + B^*)
|
|
|
|
|
| 8065128065 |
I = A A^* + B B^* + A B^* + B A^*
|
|
|
|
|
| 9761485403 |
Z
|
|
|
|
|
| 8710504862 |
A
|
|
|
|
|
| 4075539836 |
A A^* = |A|^2
|
|
|
|
|
| 6529120965 |
B
|
|
|
|
|
| 1511199318 |
Z
|
|
|
|
|
| 7107090465 |
B B^* = |B|^2
|
|
|
|
|
| 5125940051 |
I = |A|^2 + B B^* + A B^* + B A^*
|
|
|
|
|
| 1525861537 |
I = |A|^2 + |B|^2 + A B^* + B A^*
|
|
|
|
|
| 1742775076 |
Z \to B
|
|
|
|
|
| 7607271250 |
\theta \to \phi
|
|
|
|
|
| 4192519596 |
B = |B| \exp(i \phi)
|
|
|
|
|
| 2064205392 |
A
|
|
|
|
|
| 1894894315 |
Z
|
|
|
|
|
| 1357848476 |
A = |A| \exp(i \theta)
|
|
|
|
|
| 6555185548 |
A^* = |A| \exp(-i \theta)
|
|
|
|
|
| 4504256452 |
B^* = |B| \exp(-i \phi)
|
|
|
|
|
| 7621705408 |
I = |A|^2 + |B|^2 + |A| |B| \exp(-i \theta) \exp(i \phi) + |A| |B| \exp(i \theta) \exp(-i \phi)
|
|
|
|
|
| 3085575328 |
I = |A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi))
|
|
|
|
|
| 4935235303 |
x
|
|
|
|
|
| 2293352649 |
\theta - \phi
|
|
|
|
|
| 3660957533 |
\cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
|
|
|
|
|
| 3967985562 |
2
|
|
|
|
|
| 2700934933 |
2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
|
|
|
|
|
| 8497631728 |
I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )
|
|
|
|
|
| 6774684564 |
\theta = \phi
|
|
|
for coherent waves
|
|
| 2099546947 |
I = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )
|
|
|
|
|
| 2719691582 |
|A| = |B|
|
|
|
in a loop
|
|
| 3257276368 |
I = |A|^2 + |A|^2 + |A| |A| 2
|
|
|
|
|
| 8283354808 |
I_{\rm coherent} = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )
|
|
|
|
|
| 8046208134 |
I_{\rm coherent} = |A|^2 + |A|^2 + |A| |A| 2
|
|
|
|
|
| 1172039918 |
I_{\rm coherent} = 4 |A|^2
|
|
|
|
|
| 8602221482 |
\langle \cos(\theta - \phi) \rangle = 0
|
|
|
incoherent light source
|
|
| 6240206408 |
I_{\rm incoherent} = |A|^2 + |B|^2
|
|
|
|
|
| 6529793063 |
I_{\rm incoherent} = |A|^2 + |A|^2
|
|
|
|
|
| 3060393541 |
I_{\rm incoherent} = 2|A|^2
|
|
|
|
|
| 6556875579 |
\frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2
|
|
|
|
|
| 8750500372 |
f
|
|
|
|
|
| 7543102525 |
g
|
|
|
|
|
| 8267133829 |
a = b
|
|
|
|
|
| 7968822469 |
c = d
|
|
|
|
|
| 3169580383 |
\vec{a} = \frac{d\vec{v}}{dt}
|
|
|
acceleration is the change in speed over a duration
|
|
| 8602512487 |
\vec{a} = a_x \hat{x} + a_y \hat{y}
|
|
|
decompose acceleration into two components
|
|
| 4158986868 |
a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt}
|
|
|
|
|
| 5349866551 |
\vec{v} = v_x \hat{x} + v_y \hat{y}
|
|
|
|
|
| 7729413831 |
a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right)
|
|
|
|
|
| 1819663717 |
a_x = \frac{d}{dt} v_x
|
|
|
|
|
| 8228733125 |
a_y = \frac{d}{dt} v_y
|
|
|
|
|
| 9707028061 |
a_x = 0
|
|
|
|
|
| 2741489181 |
a_y = -g
|
|
|
|
|
| 3270039798 |
2
|
|
|
|
|
| 8880467139 |
2
|
|
|
|
|
| 8750379055 |
0 = \frac{d}{dt} v_x
|
|
|
|
|
| 1977955751 |
-g = \frac{d}{dt} v_y
|
|
|
|
|
| 6672141531 |
dt
|
|
|
|
|
| 1702349646 |
-g dt = d v_y
|
|
|
|
|
| 8584698994 |
-g \int dt = \int d v_y
|
|
|
|
|
| 9973952056 |
-g t = v_y - v_{0, y}
|
|
|
|
|
| 4167526462 |
v_{0, y}
|
|
|
|
|
| 6572039835 |
-g t + v_{0, y} = v_y
|
|
|
|
|
| 7252338326 |
v_y = \frac{dy}{dt}
|
|
|
|
|
| 6204539227 |
-g t + v_{0, y} = \frac{dy}{dt}
|
|
|
|
|
| 1614343171 |
dt
|
|
|
|
|
| 8145337879 |
-g t dt + v_{0, y} dt = dy
|
|
|
|
|
| 8808860551 |
-g \int t dt + v_{0, y} \int dt = \int dy
|
|
|
|
|
| 2858549874 |
- \frac{1}{2} g t^2 + v_{0, y} t = y - y_0
|
|
|
|
|
| 6098638221 |
y_0
|
|
|
|
|
| 2461349007 |
- \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y
|
|
|
|
|
| 8717193282 |
dt
|
|
|
|
|
| 1166310428 |
0 dt = d v_x
|
|
|
|
|
| 2366691988 |
\int 0 dt = \int d v_x
|
|
|
|
|
| 1676472948 |
0 = v_x - v_{0, x}
|
|
|
|
|
| 1439089569 |
v_{0, x}
|
|
|
|
|
| 6134836751 |
v_{0, x} = v_x
|
|
|
|
|
| 8460820419 |
v_x = \frac{dx}{dt}
|
|
|
|
|
| 7455581657 |
v_{0, x} = \frac{dx}{dt}
|
|
|
|
|
| 8607458157 |
dt
|
|
|
|
|
| 1963253044 |
v_{0, x} dt = dx
|
|
|
|
|
| 3676159007 |
v_{0, x} \int dt = \int dx
|
|
|
|
|
| 9882526611 |
v_{0, x} t = x - x_0
|
|
|
|
|
| 3182907803 |
x_0
|
|
|
|
|
| 8486706976 |
v_{0, x} t + x_0 = x
|
|
|
|
|
| 1306360899 |
x = v_{0, x} t + x_0
|
|
|
|
|
| 7049769409 |
2
|
|
|
|
|
| 9341391925 |
\vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y}
|
|
|
|
|
| 6410818363 |
\theta
|
|
|
|
|
| 7391837535 |
\cos(\theta) = \frac{v_{0, x}}{v_0}
|
|
|
|
|
| 7376526845 |
\sin(\theta) = \frac{v_{0, y}}{v_0}
|
|
|
|
|
| 5868731041 |
v_0
|
|
|
|
|
| 6083821265 |
v_0 \cos(\theta) = v_{0, x}
|
|
|
|
|
| 5438722682 |
x = v_0 t \cos(\theta) + x_0
|
|
|
|
|
| 5620558729 |
v_0
|
|
|
|
|
| 8949329361 |
v_0 \sin(\theta) = v_{0, y}
|
|
|
|
|
| 1405465835 |
y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0
|
|
|
|
|
| 9862900242 |
y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0
|
|
|
|
|
| 6050070428 |
v_{0, x}
|
|
|
|
|
| 3274926090 |
t = \frac{x - x_0}{v_{0, x}}
|
|
|
|
|
| 7354529102 |
y = - \frac{1}{2} g \left( \frac{x - x_0}{v_{0, x}} \right)^2 + v_{0, y} \frac{x - x_0}{v_{0, x}} + y_0
|
|
|
|
|
| 8406170337 |
y \to y_f
|
|
|
|
|
| 2403773761 |
t \to t_f
|
|
|
|
|
| 5379546684 |
y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
|
|
|
|
|
| 5373931751 |
t = t_f
|
|
|
|
|
| 9112191201 |
y_f = 0
|
|
|
|
|
| 8198310977 |
0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
|
|
|
|
|
| 1650441634 |
y_0 = 0
|
|
|
define coordinate system such that initial height is at origin
|
|
| 1087417579 |
0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)
|
|
|
|
|
| 4829590294 |
t_f
|
|
|
|
|
| 2086924031 |
0 = - \frac{1}{2} g t_f + v_0 \sin(\theta)
|
|
|
|
|
| 6974054946 |
\frac{1}{2} g t_f
|
|
|
|
|
| 1191796961 |
\frac{1}{2} g t_f = v_0 \sin(\theta)
|
|
|
|
|
| 2510804451 |
2/g
|
|
|
|
|
| 4778077984 |
t_f = \frac{2 v_0 \sin(\theta)}{g}
|
|
|
|
|
| 3273630811 |
x \to x_f
|
|
|
|
|
| 6732786762 |
t \to t_f
|
|
|
|
|
| 3485125659 |
x_f = v_0 t_f \cos(\theta) + x_0
|
|
|
|
|
| 4370074654 |
t = t_f
|
|
|
|
|
| 2378095808 |
x_f = x_0 + d
|
|
|
|
|
| 4268085801 |
x_0 + d = v_0 t_f \cos(\theta) + x_0
|
|
|
|
|
| 8072682558 |
x_0
|
|
|
|
|
| 7233558441 |
d = v_0 t_f \cos(\theta)
|
|
|
|
|
| 2297105551 |
d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)
|
|
|
|
|
| 7587034465 |
\theta
|
|
|
|
|
| 7214442790 |
x
|
|
|
|
|
| 2519058903 |
\sin(2 \theta) = 2 \sin(\theta) \cos(\theta)
|
|
|
|
|
| 8922441655 |
d = \frac{v_0^2}{g} \sin(2 \theta)
|
|
|
|
|
| 5667870149 |
\theta
|
|
|
|
|
| 1541916015 |
\theta = \frac{\pi}{4}
|
|
|
|
|
| 3607070319 |
d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)
|
|
|
|
|
| 5353282496 |
d = \frac{v_0^2}{g}
|
|
|
|
|
| 0000040490 |
a^2
|
|
|
|
|
| 0000999900 |
b/(2a)
|
|
|
|
|
| 0001030901 |
\cos(x)
|
|
|
|
|
| 0001111111 |
(\sin(x))^2
|
|
|
|
|
| 0001209482 |
2 \pi
|
|
|
|
|
| 0001304952 |
\hbar
|
|
|
|
|
| 0001334112 |
W
|
|
|
|
|
| 0001921933 |
2 i
|
|
|
|
|
| 0002239424 |
2
|
|
|
|
|
| 0002338514 |
\vec{p}_{2}
|
|
|
|
|
| 0002342425 |
m/m
|
|
|
|
|
| 0002367209 |
1/2
|
|
|
|
|
| 0002393922 |
x
|
|
|
|
|
| 0002424922 |
a
|
|
|
|
|
| 0002436656 |
i \hbar
|
|
|
|
|
| 0002449291 |
b/(2a)
|
|
|
|
|
| 0002838490 |
b/(2a)
|
|
|
|
|
| 0002919191 |
\sin(-x)
|
|
|
|
|
| 0002929944 |
1/2
|
|
|
|
|
| 0002940021 |
2 \pi
|
|
|
|
|
| 0003232242 |
t
|
|
|
|
|
| 0003413423 |
\cos(-x)
|
|
|
|
|
| 0003747849 |
-1
|
|
|
|
|
| 0003838111 |
2
|
|
|
|
|
| 0003919391 |
x
|
|
|
|
|
| 0003949052 |
-x
|
|
|
|
|
| 0003949921 |
\hbar
|
|
|
|
|
| 0003954314 |
dx
|
|
|
|
|
| 0003981813 |
-\sin(x)
|
|
|
|
|
| 0004089571 |
2x
|
|
|
|
|
| 0004264724 |
y
|
|
|
|
|
| 0004307451 |
(b/(2a))^2
|
|
|
|
|
| 0004582412 |
x
|
|
|
|
|
| 0004829194 |
2
|
|
|
|
|
| 0004831494 |
a
|
|
|
|
|
| 0004849392 |
x
|
|
|
|
|
| 0004858592 |
h
|
|
|
|
|
| 0004934845 |
x
|
|
|
|
|
| 0004948585 |
a
|
|
|
|
|
| 0005395034 |
a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
|
|
|
|
|
| 0005626421 |
t
|
|
|
|
|
| 0005749291 |
f
|
|
|
|
|
| 0005938585 |
\frac{-\hbar^2}{2m}
|
|
|
|
|
| 0006466214 |
(\sin(x))^2
|
|
|
|
|
| 0006544644 |
t
|
|
|
|
|
| 0006563727 |
x
|
|
|
|
|
| 0006644853 |
c/a
|
|
|
|
|
| 0006656532 |
e
|
|
|
|
|
| 0007471778 |
2(\sin(x))^2
|
|
|
|
|
| 0007563791 |
i
|
|
|
|
|
| 0007636749 |
x
|
|
|
|
|
| 0007894942 |
(\sin(x))^2
|
|
|
|
|
| 0008837284 |
T
|
|
|
|
|
| 0008842811 |
\cos(2x)
|
|
|
|
|
| 0009458842 |
\psi(x)
|
|
|
|
|
| 0009484724 |
\frac{n \pi}{W}x
|
|
|
|
|
| 0009485857 |
a^2\frac{2}{W}
|
|
|
|
|
| 0009485858 |
\frac{2n\pi}{W}
|
|
|
|
|
| 0009492929 |
v du
|
|
|
|
|
| 0009587738 |
\psi
|
|
|
|
|
| 0009594995 |
1/2
|
|
|
|
|
| 0009877781 |
y
|
|
|
|
|
| 0203024440 |
1 = \int_0^W a \sin(\frac{n \pi}{W} x) \psi(x)^* dx
|
|
|
|
|
| 0404050504 |
\lambda = \frac{v}{f}
|
|
|
|
|
| 0439492440 |
\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)\right|_0^W
|
|
|
|
|
| 0934990943 |
k = \frac{2 \pi}{v T}
|
|
|
|
|
| 0948572140 |
\int \cos(a x) dx = \frac{1}{a}\sin(a x)
|
|
|
|
|
| 1010393913 |
\langle \psi| \hat{A}^+ |\psi \rangle = \langle a \rangle^*
|
|
|
|
|
| 1010393944 |
x = \langle\psi_{\alpha}| a_{\beta} |\psi_{\beta} \rangle
|
|
|
|
|
| 1010923823 |
k W = n \pi
|
|
|
|
|
| 1020010291 |
0 = a \sin(k W)
|
|
|
|
|
| 1020394900 |
p = h/\lambda
|
|
|
|
|
| 1020394902 |
E = h f
|
|
|
|
|
| 1029039903 |
p = m v
|
|
|
|
|
| 1029039904 |
p^2 = m^2 v^2
|
|
|
|
|
| 1158485859 |
\frac{-\hbar^2}{2m} \nabla^2 = {\cal H}
|
|
|
|
|
| 1202310110 |
\frac{1}{a^2} = \int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
|
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| 1202312210 |
\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
|
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| 1203938249 |
a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle = a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
|
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| 1248277773 |
\cos(2x) = 1-2(\sin(x))^2
|
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| 1293913110 |
0 = b
|
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| 1293923844 |
\lambda = v T
|
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|
| 1314464131 |
\vec{\nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
|
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| 1314864131 |
\vec{\nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E}
|
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| 1395858355 |
x = \langle \psi_{\alpha}| a_{\alpha} |\psi_{\beta}\rangle
|
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| 1492811142 |
f = x - d
|
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|
| 1492842000 |
\nabla \vec{x} = f(y)
|
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|
|
| 1636453295 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = - \nabla^2 \vec{E}
|
|
|
|
|
| 1638282134 |
\vec{p}_{before} = \vec{p}_{after}
|
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|
| 1648958381 |
\nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{\nabla}\psi(\vec{r},t)\right)
|
|
|
|
|
| 1857710291 |
0 = a \sin(n \pi)
|
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|
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|
| 1858578388 |
\nabla^2 E(\vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E(\vec{r})\exp(i \omega t)
|
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|
|
|
| 1858772113 |
k = \frac{n \pi}{W}
|
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|
| 1934748140 |
\int |\psi(x)|^2 dx = 1
|
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|
|
|
| 2029293929 |
\nabla^2 E(\vec{r})\exp(i \omega t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r})\exp(i \omega t)
|
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|
|
|
| 2103023049 |
\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right)
|
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| 2113211456 |
f = 1/T
|
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|
| 2123139121 |
-\exp(-i x) = -\cos(x)+i \sin(x)
|
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| 2131616531 |
T f = 1
|
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|
| 2258485859 |
{\cal H} \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
|
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|
| 2394240499 |
x = a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle
|
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| 2394853829 |
\exp(-i x) = \cos(-x)+i \sin(-x)
|
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|
| 2394935831 |
( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
|
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|
| 2394935835 |
\left(\langle\psi| \hat{A} |\psi\rangle \right)^+ = \left(\langle a \rangle\right)^+
|
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|
| 2395958385 |
\nabla^2 \psi\left(\vec{r},t)\right) = \frac{-p^2}{\hbar} \psi(\vec{r},t)
|
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|
| 2404934990 |
\langle x^2\rangle -2\langle x \rangle\langle x \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
|
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|
| 2494533900 |
\langle x^2\rangle -\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
|
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|
| 2569154141 |
\vec{\nabla} \times \frac{\partial}{\partial t}\vec{H} = \epsilon_0 \frac{\partial^2 }{\partial t^2}\vec{E}
|
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|
| 2648958382 |
\nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p}\psi(\vec{r},t)\right)
|
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| 2848934890 |
\langle a \rangle^* = \langle a \rangle
|
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| 2944838499 |
\psi(x) = a \sin(\frac{n \pi}{W} x)
|
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|
| 3121234211 |
\frac{k}{2\pi} = \lambda
|
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|
| 3121234212 |
p = \frac{h k}{2\pi}
|
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|
| 3121513111 |
k = \frac{2 \pi}{\lambda}
|
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|
| 3131111133 |
T = 1 / f
|
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|
| 3131211131 |
\omega = 2 \pi f
|
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|
| 3132131132 |
\omega = \frac{2\pi}{T}
|
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|
| 3147472131 |
\frac{\omega}{2 \pi} = f
|
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|
| 3285732911 |
(\cos(x))^2 = 1-(\sin(x))^2
|
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|
|
| 3485475729 |
\nabla^2 E(\vec{r}) = - \frac{\omega^2}{c^2} E(\vec{r})
|
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|
| 3585845894 |
\langle \left(x-\langle x \rangle\right)^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
|
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|
| 3829492824 |
\frac{1}{2}\left(\exp(i x)+\exp(-i x)\right) = \cos(x)
|
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|
| 3924948349 |
a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
|
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|
|
| 3942849294 |
\exp(i x)-\exp(-i x) = 2 i \sin(x)
|
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|
| 3943939590 |
x = a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle
|
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|
| 3947269979 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
|
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|
| 3948571256 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi(\vec{r},t)
|
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|
|
| 3948572230 |
\vec{\nabla}\psi(\vec{r},t) = \psi_0 \vec{\nabla}\exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
|
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|
| 3948574224 |
\psi(\vec{r},t) = \psi_0 \exp\left(i\left(\vec{k}\cdot\vec{r} - \omega t \right) \right)
|
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|
|
| 3948574226 |
\psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right)
|
|
|
|
|
| 3948574228 |
\psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
|
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|
|
|
| 3948574230 |
\psi(\vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
|
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|
|
| 3948574233 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
|
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|
|
| 3948574235 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
|
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|
| 3951205425 |
\vec{p}_{after} = \vec{p}_{1}
|
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|
|
|
| 4147472132 |
E = \frac{h \omega}{2 \pi}
|
|
|
|
|
| 4298359835 |
E = \frac{1}{2}m v^2
|
|
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|
|
| 4298359845 |
E = \frac{1}{2m}m^2 v^2
|
|
|
|
|
| 4298359851 |
E = \frac{p^2}{2m}
|
|
|
|
|
| 4341171256 |
i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{p^2}{2 m} \psi(\vec{r},t)
|
|
|
|
|
| 4348571256 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi(\vec{r},t)
|
|
|
|
|
| 4394958389 |
\vec{\nabla}\cdot \left(\vec{\nabla}\psi(\vec{r},t)\right) = \frac{i}{\hbar} \vec{\nabla}\cdot\left( \vec{p} \psi(\vec{r},t)\right)
|
|
|
|
|
| 4585828572 |
\epsilon_0 \mu_0 = \frac{1}{c^2}
|
|
|
|
|
| 4585932229 |
\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
|
|
|
|
|
| 4598294821 |
\exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
|
|
|
|
|
| 4638429483 |
\exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))
|
|
|
|
|
| 4742644828 |
\exp(i x)+\exp(-i x) = 2 \cos(x)
|
|
|
|
|
| 4827492911 |
\cos(2x)+(\sin(x))^2 = 1-(\sin(x))^2
|
|
|
|
|
| 4838429483 |
\exp(2 i x) = \cos(2 x)+i \sin(2 x)
|
|
|
|
|
| 4843995999 |
\frac{1}{2 i}\left(\exp(i x)-\exp(-i x)\right) = \sin(x)
|
|
|
|
|
| 4857472413 |
1 = \int \psi(x)\psi(x)^* dx
|
|
|
|
|
| 4857475848 |
\frac{1}{a^2} = \frac{W}{2}
|
|
|
|
|
| 4858429483 |
\exp(i x)\exp(i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))
|
|
|
|
|
| 4923339482 |
i x = \log(y)
|
|
|
|
|
| 4928239482 |
\log(y) = i x
|
|
|
|
|
| 4928923942 |
a = b
|
|
|
|
|
| 4929218492 |
a+b = c
|
|
|
|
|
| 4929482992 |
b = 2
|
|
|
|
|
| 4938429482 |
\exp(-i x) = \cos(x)+i \sin(-x)
|
|
|
|
|
| 4938429483 |
\exp(i x) = \cos(x)+i \sin(x)
|
|
|
|
|
| 4938429484 |
\exp(-i x) = \cos(x)-i \sin(x)
|
|
|
|
|
| 4943571230 |
\vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
|
|
|
|
|
| 4948934890 |
\langle \psi| \hat{A} |\psi\rangle = \langle a \rangle^*
|
|
|
|
|
| 4949359835 |
\langle x^2\rangle -2\langle x^2 \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 4954839242 |
\cos(2x)+i\sin(2x) = (\cos(x)+i \sin(x))(\cos(x)+i \sin(x))
|
|
|
|
|
| 4978429483 |
\exp(-i x) = \cos(-x)+i \sin(-x)
|
|
|
|
|
| 4984892984 |
a+2 = c
|
|
|
|
|
| 4985825552 |
\nabla^2 E(\vec{r})\exp(i \omega t) = i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 5530148480 |
\vec{p}_{1}-\vec{p}_{2} = \vec{p}_{electron}
|
|
|
|
|
| 5727578862 |
\frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)
|
|
|
|
|
| 5832984291 |
(\sin(x))^2 + (\cos(x))^2 = 1
|
|
|
|
|
| 5857434758 |
\int a dx = a x
|
|
|
|
|
| 5868688585 |
\frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = \frac{p^2}{2m} \psi(\vec{r},t)
|
|
|
|
|
| 5900595848 |
k = \frac{\omega}{v}
|
|
|
|
|
| 5928285821 |
x^2 + 2x(b/(2a)) + (b/(2a))^2 = (x+(b/(2a)))^2
|
|
|
|
|
| 5928292841 |
x^2 + (b/a)x + (b/(2a))^2 = -c/a + (b/(2a))^2
|
|
|
|
|
| 5938459282 |
x^2 + (b/a)x = -c/a
|
|
|
|
|
| 5958392859 |
x^2 + (b/a)x+(c/a) = 0
|
|
|
|
|
| 5959282914 |
x^2 + x(b/a) + (b/(2a))^2 = (x+(b/(2a)))^2
|
|
|
|
|
| 5982958248 |
x = -\sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
|
|
|
|
|
| 5982958249 |
x+(b/(2a)) = -\sqrt{(b/(2a))^2 - (c/a)}
|
|
|
|
|
| 5985371230 |
\vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi(\vec{r},t)
|
|
|
|
|
| 6742123016 |
\vec{p}_{electron}\cdot\vec{p}_{electron} = (\vec{p}_{1}\cdot\vec{p}_{1})+(\vec{p}_{2}\cdot\vec{p}_{2})-2(\vec{p}_{1}\cdot\vec{p}_{2})
|
|
|
|
|
| 7466829492 |
\vec{\nabla} \cdot \vec{E} = 0
|
|
|
|
|
| 7564894985 |
\int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)
|
|
|
|
|
| 7572664728 |
\cos(2x)+2(\sin(x))^2 = 1
|
|
|
|
|
| 7575738420 |
\left(\sin\left(\frac{n \pi}{W}x\right)\right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2}
|
|
|
|
|
| 7575859295 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859300 |
\epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{\nabla} \times \vec{E} )_k = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859302 |
\epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859304 |
\epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h}
|
|
|
|
|
| 7575859306 |
\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859308 |
\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right) = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859310 |
\hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859312 |
\vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7917051060 |
\vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2}
|
|
|
|
|
| 8139187332 |
\vec{p}_{1} = \vec{p}_{2}+\vec{p}_{electron}
|
|
|
|
|
| 8257621077 |
\vec{p}_{before} = \vec{p}_{1}
|
|
|
|
|
| 8311458118 |
\vec{p}_{after} = \vec{p}_{2}+\vec{p}_{electron}
|
|
|
|
|
| 8399484849 |
\langle x^2 -2x\langle x \rangle+\langle x \rangle^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 8484544728 |
-a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x)
|
|
|
|
|
| 8485757728 |
a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(kx)
|
|
|
|
|
| 8485867742 |
\frac{2}{W} = a^2
|
|
|
|
|
| 8489593958 |
d(u v) = u dv + v du
|
|
|
|
|
| 8489593960 |
d(u v) - v du = u dv
|
|
|
|
|
| 8489593962 |
u dv = d(u v) - v du
|
|
|
|
|
| 8489593964 |
\int u dv = u v - \int v du
|
|
|
|
|
| 8494839423 |
\nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
|
|
|
|
|
| 8572657110 |
1 = \int |\psi(x)|^2 dx
|
|
|
|
|
| 8572852424 |
\vec{E} = E(\vec{r},t)
|
|
|
|
|
| 8575746378 |
\int \frac{1}{2} dx = \frac{1}{2} x
|
|
|
|
|
| 8575748999 |
\frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x)\right) = -k^2 \left(a \sin(kx) + b \cos(kx)\right)
|
|
|
|
|
| 8576785890 |
1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
|
|
|
|
|
| 8577275751 |
0 = a \sin(0) + b\cos(0)
|
|
|
|
|
| 8582885111 |
\psi(x) = a \sin(kx) + b \cos(kx)
|
|
|
|
|
| 8582954722 |
x^2 + 2xh + h^2 = (x+h)^2
|
|
|
|
|
| 8849289982 |
\psi(x)^* = a \sin(\frac{n \pi}{W} x)
|
|
|
|
|
| 8889444440 |
1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right)\right)^2 dx
|
|
|
|
|
| 9059289981 |
\psi(x) = a \sin(k x)
|
|
|
|
|
| 9285928292 |
ax^2 + bx + c = 0
|
|
|
|
|
| 9291999979 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0\vec{\nabla} \times \frac{\partial \vec{H}}{\partial t}
|
|
|
|
|
| 9294858532 |
\hat{A}^+ = \hat{A}
|
|
|
|
|
| 9385938295 |
(x+(b/(2a)))^2 = -(c/a) + (b/(2a))^2
|
|
|
|
|
| 9393939991 |
\psi(x) = -\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
|
|
|
|
|
| 9393939992 |
\psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
|
|
|
|
|
| 9394939493 |
\nabla^2 E(\vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r},t)
|
|
|
|
|
| 9429829482 |
\frac{d}{dx} y = -\sin(x) + i\cos(x)
|
|
|
|
|
| 9482113948 |
\frac{dy}{y} = i dx
|
|
|
|
|
| 9482438243 |
(\cos(x))^2 = \cos(2x)+(\sin(x))^2
|
|
|
|
|
| 9482923849 |
\exp(i x) = y
|
|
|
|
|
| 9482928242 |
\cos(2x) = (\cos(x))^2 - (\sin(x))^2
|
|
|
|
|
| 9482928243 |
\cos(2x)+(\sin(x))^2 = (\cos(x))^2
|
|
|
|
|
| 9482943948 |
\log(y) = i dx
|
|
|
|
|
| 9482948292 |
b = c
|
|
|
|
|
| 9482984922 |
\frac{d}{dx} y = (i\sin(x) + \cos(x)) i
|
|
|
|
|
| 9483928192 |
\cos(2x)+i\sin(2x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
|
|
|
|
|
| 9485384858 |
\nabla^2 E(\vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 9485747245 |
\sqrt{\frac{2}{W}} = a
|
|
|
|
|
| 9485747246 |
-\sqrt{\frac{2}{W}} = a
|
|
|
|
|
| 9492920340 |
y = \cos(x)+i \sin(x)
|
|
|
|
|
| 9494829190 |
k
|
|
|
|
|
| 9495857278 |
\psi(x=W) = 0
|
|
|
|
|
| 9499428242 |
E(\vec{r},t) = E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 9499959299 |
a + k = b + k
|
|
|
|
|
| 9582958293 |
x = \sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
|
|
|
|
|
| 9582958294 |
x+(b/(2a)) = \sqrt{(b/(2a))^2 - (c/a)}
|
|
|
|
|
| 9584821911 |
c + d = a
|
|
|
|
|
| 9585727710 |
\psi(x=0) = 0
|
|
|
|
|
| 9596004948 |
x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle
|
|
|
|
|
| 9848292229 |
dy = y i dx
|
|
|
|
|
| 9848294829 |
\frac{d}{dx} y = y i
|
|
|
|
|
| 9858028950 |
\frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
|
|
|
|
|
| 9889984281 |
2(\sin(x))^2 = 1-\cos(2x)
|
|
|
|
|
| 9919999981 |
\rho = 0
|
|
|
|
|
| 9958485859 |
\frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
|
|
|
|
|
| 9988949211 |
(\sin(x))^2 = \frac{1-\cos(2x)}{2}
|
|
|
|
|
| 9991999979 |
\vec{\nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t}
|
|
|
|
|
| 9999998870 |
\frac{\vec{p}}{\hbar} = \vec{k}
|
|
|
|
|
| 9999999870 |
\frac{p}{\hbar} = k
|
|
|
|
|
| 9999999950 |
\beta = 1/(k_{Boltzmann}T)
|
|
|
|
|
| 9999999951 |
\langle x | k \rangle = \frac{\exp(i k x)}{\sqrt{2\pi}}
|
|
|
|
|
| 9999999952 |
f(x) \delta(x-a) = f(a)
|
|
|
|
|
| 9999999953 |
\int_{-\infty}^{\infty} \delta(x) dx = 1
|
|
|
|
|
| 9999999954 |
c = 1/\sqrt{\epsilon_0 \mu_0}
|
|
|
|
|
| 9999999955 |
\vec{E} = -\vec{\nabla} \Psi
|
|
|
|
|
| 9999999956 |
\vec{F} = \frac{\partial}{\partial t}\vec{p}
|
|
|
|
|
| 9999999957 |
\vec{F} = -\vec{\nabla} V
|
|
|
|
|
| 9999999960 |
\hbar = h/(2 \pi)
|
|
|
|
|
| 9999999961 |
\frac{E}{\hbar} = \omega
|
|
|
|
|
| 9999999962 |
p = \hbar k
|
|
|
|
|
| 9999999963 |
\lambda = h/p
|
|
|
|
|
| 9999999964 |
\omega = c k
|
|
|
|
|
| 9999999965 |
E = \omega \hbar
|
|
|
|
|
| 9999999966 |
\vec{L} = \vec{r}\times\vec{p}
|
|
|
|
|
| 9999999967 |
\vec{S} = \frac{1}{\mu_0} \vec{E}\times \vec{B}
|
|
|
|
|
| 9999999968 |
x = \frac{-b-\sqrt{b^2-4ac}}{2a}
|
|
|
|
|
| 9999999969 |
x = \frac{-b+\sqrt{b^2-4ac}}{2a}
|
|
|
|
|
| 9999999970 |
\eta_1 \sin\theta_1 = \eta_2 \sin\theta_2
|
|
|
|
|
| 9999999971 |
{\cal H} = \frac{p^2}{2m}+V
|
|
|
|
|
| 9999999972 |
{\cal H} |\psi\rangle = E |\psi\rangle
|
|
|
|
|
| 9999999973 |
\left( \Delta A \right)^2 = \langle A^2 \rangle - \langle A \rangle^2
|
|
|
|
|
| 9999999974 |
\langle \psi| \hat{A} |\psi\rangle = \int_{-\infty}^{\infty} \psi^* A \psi dx
|
|
|
|
|
| 9999999975 |
\langle \psi| \hat{A} |\psi\rangle = \langle a \rangle
|
|
|
|
|
| 9999999976 |
\hat{p} = -i \hbar \frac{\partial }{\partial x}
|
|
|
|
|
| 9999999977 |
[\hat{x},\hat{p}] = i \hbar
|
|
|
|
|
| 9999999978 |
\vec{\nabla} \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial E}{\partial t}
|
|
|
|
|
| 9999999979 |
\vec{\nabla} \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}
|
|
|
|
|
| 9999999980 |
\vec{\nabla} \cdot \vec{B} = 0
|
|
|
|
|
| 9999999981 |
\vec{\nabla} \cdot \vec{E} = \rho/\epsilon_0
|
|
|
|
|
| 9999999982 |
V = I R + Q/C + L \frac{\partial I}{\partial t}
|
|
|
|
|
| 9999999983 |
C = V A/d
|
|
|
|
|
| 9999999984 |
Q = C V
|
|
|
|
|
| 9999999985 |
V = I R
|
|
|
|
|
| 9999999986 |
\left[\right]\left[\right] = \left[\right]
|
|
|
|
|
| 9999999987 |
1 atmosphere = 101.325 Pascal
|
|
|
|
|
| 9999999988 |
1 atmosphere = 14.7 pounds/(inch^2)
|
|
|
|
|
| 9999999989 |
mass_{electron} = 511000 electronVolts/(q^2)
|
|
|
|
|
| 9999999990 |
Tesla = 10000*Gauss
|
|
|
|
|
| 9999999991 |
Pascal = Newton / (meter^2)
|
|
|
|
|
| 9999999992 |
Tesla = Newton / (Ampere*meter)
|
|
|
|
|
| 9999999993 |
Farad = Coulumb / Volt
|
|
|
|
|
| 9999999995 |
Volt = Joule / Coulumb
|
|
|
|
|
| 9999999996 |
Coulomb = Ampere / second
|
|
|
|
|
| 9999999997 |
Watt = Joule / second
|
|
|
|
|
| 9999999998 |
Joule = Newton*meter
|
|
|
|
|
| 9999999999 |
Newton = kilogram*meter/(second^2)
|
|
|
|
|
| 4961662865 |
x
|
|
|
|
|
| 9110536742 |
2 x
|
|
|
|
|
| 8483686863 |
\sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x)\right)
|
|
|
|
|
| 3470587782 |
\sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
|
|
|
|
|
| 8642992037 |
2
|
|
|
|
|
| 9894826550 |
2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \left(\exp(i x)+\exp(-i x)\right)
|
|
|
|
|
| 4257214632 |
\frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)
|
|
|
|
|
| 8699789241 |
2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)
|
|
|
|
|
| 9180861128 |
2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)
|
|
|
|
|
| 2405307372 |
\sin(2 x) = 2 \sin(x) \cos(x)
|
|
|
|
|
| 3268645065 |
x
|
|
|
|
|
| 9350663581 |
\pi
|
|
|
|
|
| 8332931442 |
\exp(i \pi) = \cos(\pi)+i \sin(\pi)
|
|
|
|
|
| 6885625907 |
\exp(i \pi) = -1 + i 0
|
|
|
|
|
| 3331824625 |
\exp(i \pi) = -1
|
|
|
|
|
| 4901237716 |
1
|
|
|
|
|
| 2501591100 |
\exp(i \pi) + 1 = 0
|
|
|
|
|
| 5136652623 |
E = KE + PE
|
|
mechanical energy is the sum of the potential plus kinetic energies |
|
|
| 7915321076 |
E, KE, PE
|
|
|
|
|
| 3192374582 |
E_2, KE_2, PE_2
|
|
|
|
|
| 7875206161 |
E_2 = KE_2 + PE_2
|
|
|
|
|
| 4229092186 |
E, KE, PE
|
|
|
|
|
| 1490821948 |
E_1, KE_1, PE_1
|
|
|
|
|
| 4303372136 |
E_1 = KE_1 + PE_1
|
|
|
|
|
| 5514556106 |
E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1)
|
|
|
|
|
| 8357234146 |
KE = \frac{1}{2} m v^2
|
|
kinetic energy |
Kinetic_energy
|
|
| 3658066792 |
KE_2, v_2
|
|
|
|
|
| 8082557839 |
KE, v
|
|
|
|
|
| 7676652285 |
KE_2 = \frac{1}{2} m v_2^2
|
|
|
|
|
| 2790304597 |
KE_1, v_1
|
|
|
|
|
| 9880327189 |
KE, v
|
|
|
|
|
| 4928007622 |
KE_1 = \frac{1}{2} m v_1^2
|
|
|
|
|
| 5733146966 |
KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right)
|
|
|
|
|
| 6554292307 |
t
|
|
|
|
|
| 4270680309 |
\frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t}
|
|
|
|
|
| 5781981178 |
x^2 - y^2 = (x+y)(x-y)
|
|
difference of squares |
Difference_of_two_squares
|
|
| 5946759357 |
v_2, v_1
|
|
|
|
|
| 6675701064 |
x, y
|
|
|
|
|
| 4648451961 |
v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1)
|
|
|
|
|
| 9356924046 |
\frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t}
|
|
|
|
|
| 9397152918 |
v = \frac{v_1 + v_2}{2}
|
|
average velocity |
|
|
| 2857430695 |
a = \frac{v_2 - v_1}{t}
|
|
acceleration |
|
|
| 7735737409 |
\frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t}
|
|
|
|
|
| 4784793837 |
\frac{KE_2 - KE_1}{t} = m v a
|
|
|
|
|
| 5345738321 |
F = m a
|
|
Newton's second law of motion |
Newton%27s_laws_of_motion#Newton's_second_law
|
|
| 2186083170 |
\frac{KE_2 - KE_1}{t} = v F
|
|
|
|
|
| 6715248283 |
PE = -F x
|
|
potential energy |
Potential_energy
|
|
| 3852078149 |
PE_2, x_2
|
|
|
|
|
| 7388921188 |
PE, x
|
|
|
|
|
| 2431507955 |
PE_2 = -F x_2
|
|
|
|
|
| 3412641898 |
PE_1, x_1
|
|
|
|
|
| 9937169749 |
PE, x
|
|
|
|
|
| 4669290568 |
PE_1 = -F x_1
|
|
|
|
|
| 7734996511 |
PE_2 - PE_1 = -F ( x_2 - x_1 )
|
|
|
|
|
| 2016063530 |
t
|
|
|
|
|
| 7267155233 |
\frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right)
|
|
|
|
|
| 9337785146 |
v = \frac{x_2 - x_1}{t}
|
|
average velocity |
|
|
| 4872970974 |
\frac{PE_2 - PE_1}{t} = -F v
|
|
|
|
|
| 2081689540 |
t
|
|
|
|
|
| 2770069250 |
\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t}
|
|
|
|
|
| 3591237106 |
\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v
|
|
|
|
|
| 1772416655 |
\frac{E_2 - E_1}{t} = v F - F v
|
|
|
|
|
| 1809909100 |
\frac{E_2 - E_1}{t} = 0
|
|
|
|
|
| 5778176146 |
t
|
|
|
|
|
| 3806977900 |
E_2 - E_1 = 0
|
|
|
|
|
| 5960438249 |
E_1
|
|
|
|
|
| 8558338742 |
E_2 = E_1
|
|
|
|
|
| 8945218208 |
\theta_{Brewster} + \theta_{refracted} = 90^{\circ}
|
|
|
based on figure 34-27 on page 824 in 2001_HRW
|
|
| 9025853427 |
\theta_{Brewster}
|
|
|
|
|
| 1310571337 |
\theta_{refracted} = 90^{\circ} - \theta_{Brewster}
|
|
|
|
|
| 6450985774 |
n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 )
|
|
Law of Refraction |
eq 34-44 on page 819 in 2001_HRW
|
|
| 1779497048 |
\theta_{Brewster}, \theta_{refraction}
|
|
|
|
|
| 7322344455 |
\theta_1, \theta_2
|
|
|
|
|
| 2575937347 |
n_1 \sin( \theta_{Brewster} ) = n_2 \sin( \theta_{refraction} )
|
|
|
|
|
| 7696214507 |
n_1 \sin( \theta_{Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{Brewster} )
|
|
|
|
|
| 8588429722 |
\sin( 90^{\circ} - x ) = \cos( x )
|
|
|
|
|
| 7375348852 |
\theta_{Brewster}
|
|
|
|
|
| 1512581563 |
x
|
|
|
|
|
| 6831637424 |
\sin( 90^{\circ} - \theta_{Brewster} ) = \cos( \theta_{Brewster} )
|
|
|
|
|
| 3061811650 |
n_1 \sin( \theta_{Brewster} ) = n_2 \cos( \theta_{Brewster} )
|
|
|
|
|
| 7857757625 |
n_1
|
|
|
|
|
| 9756089533 |
\sin( \theta_{Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{Brewster} )
|
|
|
|
|
| 5632428182 |
\cos( \theta_{Brewster} )
|
|
|
|
|
| 2768857871 |
\frac{\sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )} = \frac{n_2}{n_1}
|
|
|
|
|
| 4968680693 |
\tan( x ) = \frac{ \sin( x )}{\cos( x )}
|
|
|
|
|
| 7321695558 |
\theta_{Brewster}
|
|
|
|
|
| 9906920183 |
x
|
|
|
|
|
| 4501377629 |
\tan( \theta_{Brewster} ) = \frac{ \sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )}
|
|
|
|
|
| 3417126140 |
\tan( \theta_{Brewster} ) = \frac{ n_2 }{ n_1 }
|
|
|
|
|
| 5453995431 |
\arctan{ x }
|
|
|
|
|
| 6023986360 |
x
|
|
|
|
|
| 8495187962 |
\theta_{Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }
|
|
|
|
|
| 9040079362 |
f
|
|
|
|
|
| 9565166889 |
T
|
|
|
|
|
| 5426308937 |
v = \frac{d}{t}
|
|
|
|
|
| 7476820482 |
C
|
|
|
|
|
| 1277713901 |
d
|
|
|
|
|
| 6946088325 |
v = \frac{C}{t}
|
|
|
|
|
| 6785303857 |
C = 2 \pi r
|
|
|
|
|
| 4816195267 |
C, r
|
|
|
|
|
| 2113447367 |
C_{\rm{Earth\ orbit}}, r_{\rm{Earth\ orbit}}
|
|
|
|
|
| 6348260313 |
C_{\rm{Earth\ orbit}} = 2 \pi r_{\rm{Earth\ orbit}}
|
|
|
|
|
| 3847777611 |
C, v, t
|
|
|
|
|
| 6406487188 |
C_{\rm{Earth\ orbit}}, v_{\rm{Earth\ orbit}}, t_{\rm{Earth\ orbit}}
|
|
|
|
|
| 3046191961 |
v_{\rm{Earth\ orbit}} = \frac{C_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
|
|
|
|
|
| 3080027960 |
v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
|
|
|
|
|
| 7175416299 |
t_{\rm{Earth\ orbit}} = 1 {\rm{year}}
|
|
|
|
|
| 3219318145 |
\frac{365 {\rm days}}{1 {\rm year}} \frac{24 {\rm hours}}{1 {\rm day}} \frac{60 {\rm minutes}}{1 {\rm hour}} \frac{60 {\rm seconds}}{1 {\rm minute}}
|
|
|
|
|
| 8721295221 |
t_{\rm{Earth\ orbit}} = 3.16 10^7 {\rm{seconds}}
|
|
|
|
|
| 4593428198 |
v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{3.16\ 10^7 {\rm seconds}}
|
|
|
|
|
| 3472836147 |
r_{\rm{Earth\ orbit}} = 1.496\ 10^8 {\rm km}
|
|
|
|
|
| 6998364753 |
v_{\rm{Earth\ orbit}} = \frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}}
|
|
|
|
|
| 4180845508 |
v_{\rm{Earth\ orbit}} = 29.8 \frac{{\rm km}}{{\rm sec}}
|
|
|
|
|
| 4662369843 |
x' = \gamma (x - v t)
|
|
|
|
|
| 2983053062 |
x = \gamma (x' + v t')
|
|
|
|
|
| 3426941928 |
x = \gamma ( \gamma (x - v t) + v t' )
|
|
|
|
|
| 2096918413 |
x = \gamma ( \gamma x - \gamma v t + v t' )
|
|
|
|
|
| 7741202861 |
x = \gamma^2 x - \gamma^2 v t + \gamma v t'
|
|
|
|
|
| 7337056406 |
\gamma^2 x
|
|
|
|
|
| 4139999399 |
x - \gamma^2 x = - \gamma^2 v t + \gamma v t'
|
|
|
|
|
| 3495403335 |
x
|
|
|
|
|
| 9031609275 |
x (1 - \gamma^2 ) = - \gamma^2 v t + \gamma v t'
|
|
|
|
|
| 8014566709 |
\gamma^2 v t
|
|
|
|
|
| 9409776983 |
x (1 - \gamma^2 ) + \gamma^2 v t = \gamma v t'
|
|
|
|
|
| 2226340358 |
\gamma v
|
|
|
|
|
| 6417705759 |
\frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = \gamma v t'
|
|
|
|
|
| 8730201316 |
\frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t = t'
|
|
|
first term was multiplied by \gamma/\gamma
|
|
| 1974334644 |
\frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = t'
|
|
|
|
|
| 5148266645 |
t' = \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t
|
|
|
|
|
| 4287102261 |
x^2 + y^2 + z^2 = c^2 t^2
|
|
|
describes a spherical wavefront
|
|
| 1201689765 |
x'^2 + y'^2 + z'^2 = c^2 t'^2
|
|
|
describes a spherical wavefront for an observer in a moving frame of reference
|
|
| 7057864873 |
y' = y
|
|
|
frame of reference is moving only along x direction
|
|
| 8515803375 |
z' = z
|
|
|
frame of reference is moving only along x direction
|
|
| 6153463771 |
3
|
|
|
|
|
| 4746576736 |
asdf
|
|
|
|
|
| 1027270623 |
minga
|
|
|
|
|
| 6101803533 |
ming
|
|
|
|
|
| 9231495607 |
a = b
|
|
|
|
|
| 8664264194 |
b + 2
|
|
|
|
|
| 9805063945 |
\gamma^2 (x - v t)^2 + y^2 + z^2 = c^2 \gamma^2 \left( t + \frac{ 1 - \gamma^2 }{ \gamma^2 } \frac{x}{v} \right)^2
|
|
|
|
|
| 4057686137 |
C \to C_{\rm{Earth\ orbit}}
|
|
|
|
|
| 2346150725 |
r \to r_{\rm{Earth\ orbit}}
|
|
|
|
|
| 9753878784 |
v \to v_{\rm{Earth\ orbit}}
|
|
|
|
|
| 8135396036 |
t \to t_{\rm{Earth\ orbit}}
|
|
|
|
|
| 2773628333 |
\theta_1 \to \theta_{Brewster}
|
|
|
|
|
| 7154592211 |
\theta_2 \to \theta_{\rm refracted}
|
|
|
|
|
| 3749492596 |
E \to E_1
|
|
|
|
|
| 1258245373 |
E \to E_2
|
|
|
|
|
| 4147101187 |
KE \to KE_1
|
|
|
|
|
| 3809726424 |
PE \to PE_1
|
|
|
|
|
| 6383056612 |
KE \to KE_2
|
|
|
|
|
| 5075406409 |
PE \to PE_2
|
|
|
|
|
| 5398681503 |
v \to v_1
|
|
|
|
|
| 9305761407 |
v \to v_2
|
|
|
|
|
| 4218009993 |
x \to x_1
|
|
|
|
|
| 4188639044 |
x \to x_2
|
|
|
|
|
| 8173074178 |
x \to v_2
|
|
|
|
|
| 1025759423 |
y \to v_1
|
|
|
|
|
| 1935543849 |
\gamma^2 x^2 - \gamma^2 2 x v t + \gamma^2 v^2 t^2 + y^2 + z^2 = c^2 \gamma^2 \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x^2}{\gamma^2} + c^2 \gamma^2 2 t \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x}{\gamma} + c^2 \gamma^2 t^2
|
|
|
|
|
| 1586866563 |
\left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right) = t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)
|
|
|
|
|
| 3182633789 |
\gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1
|
|
|
|
|
| 1916173354 |
-\gamma^2 v^2 + c^2 \gamma^2 = c^2
|
|
|
|
|
| 2076171250 |
-\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} = 0
|
|
|
|
|
| 5284610349 |
\gamma^2
|
|
|
|
|
| 2417941373 |
- c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 - \gamma^2
|
|
|
|
|
| 5787469164 |
1 - \gamma^2
|
|
|
|
|
| 1639827492 |
- c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2} = 1
|
|
|
|
|
| 5669500954 |
v^2 \gamma^2
|
|
|
|
|
| 5763749235 |
-c^2 + c^2 \gamma^2 = v^2 \gamma^2
|
|
|
|
|
| 6408214498 |
c^2
|
|
|
|
|
| 2999795755 |
c^2 \gamma^2 = v^2 \gamma^2 + c^2
|
|
|
|
|
| 3412946408 |
v^2 \gamma^2
|
|
|
|
|
| 2542420160 |
c^2 \gamma^2 - v^2 \gamma^2 = c^2
|
|
|
|
|
| 7743841045 |
\gamma^2
|
|
|
|
|
| 7513513483 |
\gamma^2 (c^2 - v^2) = c^2
|
|
|
|
|
| 8571466509 |
c^2 - \gamma^2
|
|
|
|
|
| 7906112355 |
\gamma^2 = \frac{c^2}{c^2 - \gamma^2}
|
|
|
|
|
| 3060139639 |
1/2
|
|
|
|
|
| 1528310784 |
\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
|
|
|
|
|
| 8360117126 |
\gamma = \frac{-1}{\sqrt{1-\frac{v^2}{c^2}}}
|
|
|
not a physically valid result in this context
|
|
| 3366703541 |
a = \frac{v - v_0}{t}
|
|
|
acceleration is the average change in speed over a duration
|
|
| 7083390553 |
t
|
|
|
|
|
| 4748157455 |
a t = v - v_0
|
|
|
|
|
| 6417359412 |
v_0
|
|
|
|
|
| 4798787814 |
a t + v_0 = v
|
|
|
|
|
| 3462972452 |
v = v_0 + a t
|
|
|
|
|
| 3411994811 |
v_{\rm ave} = \frac{d}{t}
|
|
|
|
|
| 6175547907 |
v_{\rm ave} = \frac{v + v_0}{2}
|
|
|
|
|
| 9897284307 |
\frac{d}{t} = \frac{v + v_0}{2}
|
|
|
|
|
| 8865085668 |
t
|
|
|
|
|
| 8706092970 |
d = \left(\frac{v + v_0}{2}\right)t
|
|
|
|
|
| 7011114072 |
d = \frac{(v_0 + a t) + v_0}{2} t
|
|
|
|
|
| 1265150401 |
d = \frac{2 v_0 + a t}{2} t
|
|
|
|
|
| 9658195023 |
d = v_0 t + \frac{1}{2} a t^2
|
|
|
|
|
| 5799753649 |
2
|
|
|
|
|
| 7215099603 |
v^2 = v_0^2 + 2 a t v_0 + a^2 t^2
|
|
|
|
|
| 5144263777 |
v^2 = v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right)
|
|
|
|
|
| 7939765107 |
v^2 = v_0^2 + 2 a d
|
|
|
|
|
| 6729698807 |
v_0
|
|
|
|
|
| 9759901995 |
v - v_0 = a t
|
|
|
|
|
| 2242144313 |
a
|
|
|
|
|
| 1967582749 |
t = \frac{v - v_0}{a}
|
|
|
|
|
| 5733721198 |
d = \frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right)
|
|
|
|
|
| 5611024898 |
d = \frac{1}{2 a} (v^2 - v_0^2)
|
|
|
|
|
| 5542390646 |
2 a
|
|
|
|
|
| 8269198922 |
2 a d = v^2 - v_0^2
|
|
|
|
|
| 9070454719 |
v_0^2
|
|
|
|
|
| 4948763856 |
2 a d + v_0^2 = v^2
|
|
|
|
|
| 9645178657 |
a t
|
|
|
|
|
| 6457044853 |
v - a t = v_0
|
|
|
|
|
| 1259826355 |
d = (v - a t) t + \frac{1}{2} a t^2
|
|
|
|
|
| 4580545876 |
d = v t - a t^2 + \frac{1}{2} a t^2
|
|
|
|
|
| 6421241247 |
d = v t - \frac{1}{2} a t^2
|
|
|
|
|
| 4182362050 |
Z = |Z| \exp( i \theta )
|
|
|
Z \in \mathbb{C}
|
|
| 1928085940 |
Z^* = |Z| \exp( -i \theta )
|
|
|
|
|
| 9191880568 |
Z Z^* = |Z| |Z| \exp( -i \theta ) \exp( i \theta )
|
|
|
|
|
| 3350830826 |
Z Z^* = |Z|^2
|
|
|
|
|
| 8396997949 |
I = | A + B |^2
|
|
|
intensity of two waves traveling opposite directions on same path
|
|
| 4437214608 |
Z
|
|
|
|
|
| 5623794884 |
A + B
|
|
|
|
|
| 2236639474 |
(A + B)(A + B)^* = |A + B|^2
|
|
|
|
|
| 1020854560 |
I = (A + B)(A + B)^*
|
|
|
|
|
| 6306552185 |
I = (A + B)(A^* + B^*)
|
|
|
|
|
| 8065128065 |
I = A A^* + B B^* + A B^* + B A^*
|
|
|
|
|
| 9761485403 |
Z
|
|
|
|
|
| 8710504862 |
A
|
|
|
|
|
| 4075539836 |
A A^* = |A|^2
|
|
|
|
|
| 6529120965 |
B
|
|
|
|
|
| 1511199318 |
Z
|
|
|
|
|
| 7107090465 |
B B^* = |B|^2
|
|
|
|
|
| 5125940051 |
I = |A|^2 + B B^* + A B^* + B A^*
|
|
|
|
|
| 1525861537 |
I = |A|^2 + |B|^2 + A B^* + B A^*
|
|
|
|
|
| 1742775076 |
Z \to B
|
|
|
|
|
| 7607271250 |
\theta \to \phi
|
|
|
|
|
| 4192519596 |
B = |B| \exp(i \phi)
|
|
|
|
|
| 2064205392 |
A
|
|
|
|
|
| 1894894315 |
Z
|
|
|
|
|
| 1357848476 |
A = |A| \exp(i \theta)
|
|
|
|
|
| 6555185548 |
A^* = |A| \exp(-i \theta)
|
|
|
|
|
| 4504256452 |
B^* = |B| \exp(-i \phi)
|
|
|
|
|
| 7621705408 |
I = |A|^2 + |B|^2 + |A| |B| \exp(-i \theta) \exp(i \phi) + |A| |B| \exp(i \theta) \exp(-i \phi)
|
|
|
|
|
| 3085575328 |
I = |A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi))
|
|
|
|
|
| 4935235303 |
x
|
|
|
|
|
| 2293352649 |
\theta - \phi
|
|
|
|
|
| 3660957533 |
\cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
|
|
|
|
|
| 3967985562 |
2
|
|
|
|
|
| 2700934933 |
2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
|
|
|
|
|
| 8497631728 |
I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )
|
|
|
|
|
| 6774684564 |
\theta = \phi
|
|
|
for coherent waves
|
|
| 2099546947 |
I = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )
|
|
|
|
|
| 2719691582 |
|A| = |B|
|
|
|
in a loop
|
|
| 3257276368 |
I = |A|^2 + |A|^2 + |A| |A| 2
|
|
|
|
|
| 8283354808 |
I_{\rm coherent} = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )
|
|
|
|
|
| 8046208134 |
I_{\rm coherent} = |A|^2 + |A|^2 + |A| |A| 2
|
|
|
|
|
| 1172039918 |
I_{\rm coherent} = 4 |A|^2
|
|
|
|
|
| 8602221482 |
\langle \cos(\theta - \phi) \rangle = 0
|
|
|
incoherent light source
|
|
| 6240206408 |
I_{\rm incoherent} = |A|^2 + |B|^2
|
|
|
|
|
| 6529793063 |
I_{\rm incoherent} = |A|^2 + |A|^2
|
|
|
|
|
| 3060393541 |
I_{\rm incoherent} = 2|A|^2
|
|
|
|
|
| 6556875579 |
\frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2
|
|
|
|
|
| 8750500372 |
f
|
|
|
|
|
| 7543102525 |
g
|
|
|
|
|
| 8267133829 |
a = b
|
|
|
|
|
| 7968822469 |
c = d
|
|
|
|
|
| 3169580383 |
\vec{a} = \frac{d\vec{v}}{dt}
|
|
|
acceleration is the change in speed over a duration
|
|
| 8602512487 |
\vec{a} = a_x \hat{x} + a_y \hat{y}
|
|
|
decompose acceleration into two components
|
|
| 4158986868 |
a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt}
|
|
|
|
|
| 5349866551 |
\vec{v} = v_x \hat{x} + v_y \hat{y}
|
|
|
|
|
| 7729413831 |
a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right)
|
|
|
|
|
| 1819663717 |
a_x = \frac{d}{dt} v_x
|
|
|
|
|
| 8228733125 |
a_y = \frac{d}{dt} v_y
|
|
|
|
|
| 9707028061 |
a_x = 0
|
|
|
|
|
| 2741489181 |
a_y = -g
|
|
|
|
|
| 3270039798 |
2
|
|
|
|
|
| 8880467139 |
2
|
|
|
|
|
| 8750379055 |
0 = \frac{d}{dt} v_x
|
|
|
|
|
| 1977955751 |
-g = \frac{d}{dt} v_y
|
|
|
|
|
| 6672141531 |
dt
|
|
|
|
|
| 1702349646 |
-g dt = d v_y
|
|
|
|
|
| 8584698994 |
-g \int dt = \int d v_y
|
|
|
|
|
| 9973952056 |
-g t = v_y - v_{0, y}
|
|
|
|
|
| 4167526462 |
v_{0, y}
|
|
|
|
|
| 6572039835 |
-g t + v_{0, y} = v_y
|
|
|
|
|
| 7252338326 |
v_y = \frac{dy}{dt}
|
|
|
|
|
| 6204539227 |
-g t + v_{0, y} = \frac{dy}{dt}
|
|
|
|
|
| 1614343171 |
dt
|
|
|
|
|
| 8145337879 |
-g t dt + v_{0, y} dt = dy
|
|
|
|
|
| 8808860551 |
-g \int t dt + v_{0, y} \int dt = \int dy
|
|
|
|
|
| 2858549874 |
- \frac{1}{2} g t^2 + v_{0, y} t = y - y_0
|
|
|
|
|
| 6098638221 |
y_0
|
|
|
|
|
| 2461349007 |
- \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y
|
|
|
|
|
| 8717193282 |
dt
|
|
|
|
|
| 1166310428 |
0 dt = d v_x
|
|
|
|
|
| 2366691988 |
\int 0 dt = \int d v_x
|
|
|
|
|
| 1676472948 |
0 = v_x - v_{0, x}
|
|
|
|
|
| 1439089569 |
v_{0, x}
|
|
|
|
|
| 6134836751 |
v_{0, x} = v_x
|
|
|
|
|
| 8460820419 |
v_x = \frac{dx}{dt}
|
|
|
|
|
| 7455581657 |
v_{0, x} = \frac{dx}{dt}
|
|
|
|
|
| 8607458157 |
dt
|
|
|
|
|
| 1963253044 |
v_{0, x} dt = dx
|
|
|
|
|
| 3676159007 |
v_{0, x} \int dt = \int dx
|
|
|
|
|
| 9882526611 |
v_{0, x} t = x - x_0
|
|
|
|
|
| 3182907803 |
x_0
|
|
|
|
|
| 8486706976 |
v_{0, x} t + x_0 = x
|
|
|
|
|
| 1306360899 |
x = v_{0, x} t + x_0
|
|
|
|
|
| 7049769409 |
2
|
|
|
|
|
| 9341391925 |
\vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y}
|
|
|
|
|
| 6410818363 |
\theta
|
|
|
|
|
| 7391837535 |
\cos(\theta) = \frac{v_{0, x}}{v_0}
|
|
|
|
|
| 7376526845 |
\sin(\theta) = \frac{v_{0, y}}{v_0}
|
|
|
|
|
| 5868731041 |
v_0
|
|
|
|
|
| 6083821265 |
v_0 \cos(\theta) = v_{0, x}
|
|
|
|
|
| 5438722682 |
x = v_0 t \cos(\theta) + x_0
|
|
|
|
|
| 5620558729 |
v_0
|
|
|
|
|
| 8949329361 |
v_0 \sin(\theta) = v_{0, y}
|
|
|
|
|
| 1405465835 |
y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0
|
|
|
|
|
| 9862900242 |
y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0
|
|
|
|
|
| 6050070428 |
v_{0, x}
|
|
|
|
|
| 3274926090 |
t = \frac{x - x_0}{v_{0, x}}
|
|
|
|
|
| 7354529102 |
y = - \frac{1}{2} g \left( \frac{x - x_0}{v_{0, x}} \right)^2 + v_{0, y} \frac{x - x_0}{v_{0, x}} + y_0
|
|
|
|
|
| 8406170337 |
y \to y_f
|
|
|
|
|
| 2403773761 |
t \to t_f
|
|
|
|
|
| 5379546684 |
y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
|
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| 5373931751 |
t = t_f
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| 9112191201 |
y_f = 0
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| 8198310977 |
0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
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| 1650441634 |
y_0 = 0
|
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|
define coordinate system such that initial height is at origin
|
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| 1087417579 |
0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)
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| 4829590294 |
t_f
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| 2086924031 |
0 = - \frac{1}{2} g t_f + v_0 \sin(\theta)
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| 6974054946 |
\frac{1}{2} g t_f
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| 1191796961 |
\frac{1}{2} g t_f = v_0 \sin(\theta)
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| 2510804451 |
2/g
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| 4778077984 |
t_f = \frac{2 v_0 \sin(\theta)}{g}
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| 3273630811 |
x \to x_f
|
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| 6732786762 |
t \to t_f
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| 3485125659 |
x_f = v_0 t_f \cos(\theta) + x_0
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| 4370074654 |
t = t_f
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| 2378095808 |
x_f = x_0 + d
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| 4268085801 |
x_0 + d = v_0 t_f \cos(\theta) + x_0
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| 8072682558 |
x_0
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| 7233558441 |
d = v_0 t_f \cos(\theta)
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| 2297105551 |
d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)
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| 7587034465 |
\theta
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| 7214442790 |
x
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| 2519058903 |
\sin(2 \theta) = 2 \sin(\theta) \cos(\theta)
|
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| 8922441655 |
d = \frac{v_0^2}{g} \sin(2 \theta)
|
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|
| 5667870149 |
\theta
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| 1541916015 |
\theta = \frac{\pi}{4}
|
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| 3607070319 |
d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)
|
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|
| 5353282496 |
d = \frac{v_0^2}{g}
|
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|
| 0000040490 |
a^2
|
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|
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| 0000999900 |
b/(2a)
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| 0001030901 |
\cos(x)
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| 0001111111 |
(\sin(x))^2
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| 0001209482 |
2 \pi
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| 0001304952 |
\hbar
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| 0001334112 |
W
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| 0001921933 |
2 i
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| 0002239424 |
2
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| 0002338514 |
\vec{p}_{2}
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| 0002342425 |
m/m
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| 0002367209 |
1/2
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| 0002393922 |
x
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| 0002424922 |
a
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| 0002436656 |
i \hbar
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| 0002449291 |
b/(2a)
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| 0002838490 |
b/(2a)
|
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| 0002919191 |
\sin(-x)
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| 0002929944 |
1/2
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| 0002940021 |
2 \pi
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| 0003232242 |
t
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| 0003413423 |
\cos(-x)
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| 0003747849 |
-1
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| 0003838111 |
2
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| 0003919391 |
x
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| 0003949052 |
-x
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| 0003949921 |
\hbar
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| 0003954314 |
dx
|
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| 0003981813 |
-\sin(x)
|
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| 0004089571 |
2x
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| 0004264724 |
y
|
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| 0004307451 |
(b/(2a))^2
|
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| 0004582412 |
x
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| 0004829194 |
2
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| 0004831494 |
a
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| 0004849392 |
x
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| 0004858592 |
h
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| 0004934845 |
x
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| 0004948585 |
a
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| 0005395034 |
a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
|
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| 0005626421 |
t
|
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| 0005749291 |
f
|
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|
| 0005938585 |
\frac{-\hbar^2}{2m}
|
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|
|
| 0006466214 |
(\sin(x))^2
|
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|
| 0006544644 |
t
|
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| 0006563727 |
x
|
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| 0006644853 |
c/a
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| 0006656532 |
e
|
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| 0007471778 |
2(\sin(x))^2
|
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| 0007563791 |
i
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| 0007636749 |
x
|
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| 0007894942 |
(\sin(x))^2
|
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|
| 0008837284 |
T
|
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|
|
| 0008842811 |
\cos(2x)
|
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|
|
|
| 0009458842 |
\psi(x)
|
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|
| 0009484724 |
\frac{n \pi}{W}x
|
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|
|
| 0009485857 |
a^2\frac{2}{W}
|
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|
|
|
| 0009485858 |
\frac{2n\pi}{W}
|
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|
|
| 0009492929 |
v du
|
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|
|
| 0009587738 |
\psi
|
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|
|
| 0009594995 |
1/2
|
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|
|
| 0009877781 |
y
|
|
|
|
|
| 0203024440 |
1 = \int_0^W a \sin(\frac{n \pi}{W} x) \psi(x)^* dx
|
|
|
|
|
| 0404050504 |
\lambda = \frac{v}{f}
|
|
|
|
|
| 0439492440 |
\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)\right|_0^W
|
|
|
|
|
| 0934990943 |
k = \frac{2 \pi}{v T}
|
|
|
|
|
| 0948572140 |
\int \cos(a x) dx = \frac{1}{a}\sin(a x)
|
|
|
|
|
| 1010393913 |
\langle \psi| \hat{A}^+ |\psi \rangle = \langle a \rangle^*
|
|
|
|
|
| 1010393944 |
x = \langle\psi_{\alpha}| a_{\beta} |\psi_{\beta} \rangle
|
|
|
|
|
| 1010923823 |
k W = n \pi
|
|
|
|
|
| 1020010291 |
0 = a \sin(k W)
|
|
|
|
|
| 1020394900 |
p = h/\lambda
|
|
|
|
|
| 1020394902 |
E = h f
|
|
|
|
|
| 1029039903 |
p = m v
|
|
|
|
|
| 1029039904 |
p^2 = m^2 v^2
|
|
|
|
|
| 1158485859 |
\frac{-\hbar^2}{2m} \nabla^2 = {\cal H}
|
|
|
|
|
| 1202310110 |
\frac{1}{a^2} = \int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
|
|
|
|
|
| 1202312210 |
\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
|
|
|
|
|
| 1203938249 |
a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle = a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
|
|
|
|
|
| 1248277773 |
\cos(2x) = 1-2(\sin(x))^2
|
|
|
|
|
| 1293913110 |
0 = b
|
|
|
|
|
| 1293923844 |
\lambda = v T
|
|
|
|
|
| 1314464131 |
\vec{\nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
|
|
|
|
|
| 1314864131 |
\vec{\nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E}
|
|
|
|
|
| 1395858355 |
x = \langle \psi_{\alpha}| a_{\alpha} |\psi_{\beta}\rangle
|
|
|
|
|
| 1492811142 |
f = x - d
|
|
|
|
|
| 1492842000 |
\nabla \vec{x} = f(y)
|
|
|
|
|
| 1636453295 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = - \nabla^2 \vec{E}
|
|
|
|
|
| 1638282134 |
\vec{p}_{before} = \vec{p}_{after}
|
|
|
|
|
| 1648958381 |
\nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{\nabla}\psi(\vec{r},t)\right)
|
|
|
|
|
| 1857710291 |
0 = a \sin(n \pi)
|
|
|
|
|
| 1858578388 |
\nabla^2 E(\vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 1858772113 |
k = \frac{n \pi}{W}
|
|
|
|
|
| 1934748140 |
\int |\psi(x)|^2 dx = 1
|
|
|
|
|
| 2029293929 |
\nabla^2 E(\vec{r})\exp(i \omega t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 2103023049 |
\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right)
|
|
|
|
|
| 2113211456 |
f = 1/T
|
|
|
|
|
| 2123139121 |
-\exp(-i x) = -\cos(x)+i \sin(x)
|
|
|
|
|
| 2131616531 |
T f = 1
|
|
|
|
|
| 2258485859 |
{\cal H} \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
|
|
|
|
|
| 2394240499 |
x = a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle
|
|
|
|
|
| 2394853829 |
\exp(-i x) = \cos(-x)+i \sin(-x)
|
|
|
|
|
| 2394935831 |
( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
|
|
|
|
|
| 2394935835 |
\left(\langle\psi| \hat{A} |\psi\rangle \right)^+ = \left(\langle a \rangle\right)^+
|
|
|
|
|
| 2395958385 |
\nabla^2 \psi\left(\vec{r},t)\right) = \frac{-p^2}{\hbar} \psi(\vec{r},t)
|
|
|
|
|
| 2404934990 |
\langle x^2\rangle -2\langle x \rangle\langle x \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 2494533900 |
\langle x^2\rangle -\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 2569154141 |
\vec{\nabla} \times \frac{\partial}{\partial t}\vec{H} = \epsilon_0 \frac{\partial^2 }{\partial t^2}\vec{E}
|
|
|
|
|
| 2648958382 |
\nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p}\psi(\vec{r},t)\right)
|
|
|
|
|
| 2848934890 |
\langle a \rangle^* = \langle a \rangle
|
|
|
|
|
| 2944838499 |
\psi(x) = a \sin(\frac{n \pi}{W} x)
|
|
|
|
|
| 3121234211 |
\frac{k}{2\pi} = \lambda
|
|
|
|
|
| 3121234212 |
p = \frac{h k}{2\pi}
|
|
|
|
|
| 3121513111 |
k = \frac{2 \pi}{\lambda}
|
|
|
|
|
| 3131111133 |
T = 1 / f
|
|
|
|
|
| 3131211131 |
\omega = 2 \pi f
|
|
|
|
|
| 3132131132 |
\omega = \frac{2\pi}{T}
|
|
|
|
|
| 3147472131 |
\frac{\omega}{2 \pi} = f
|
|
|
|
|
| 3285732911 |
(\cos(x))^2 = 1-(\sin(x))^2
|
|
|
|
|
| 3485475729 |
\nabla^2 E(\vec{r}) = - \frac{\omega^2}{c^2} E(\vec{r})
|
|
|
|
|
| 3585845894 |
\langle \left(x-\langle x \rangle\right)^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 3829492824 |
\frac{1}{2}\left(\exp(i x)+\exp(-i x)\right) = \cos(x)
|
|
|
|
|
| 3924948349 |
a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
|
|
|
|
|
| 3942849294 |
\exp(i x)-\exp(-i x) = 2 i \sin(x)
|
|
|
|
|
| 3943939590 |
x = a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle
|
|
|
|
|
| 3947269979 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
|
|
|
|
|
| 3948571256 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi(\vec{r},t)
|
|
|
|
|
| 3948572230 |
\vec{\nabla}\psi(\vec{r},t) = \psi_0 \vec{\nabla}\exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
|
|
|
|
|
| 3948574224 |
\psi(\vec{r},t) = \psi_0 \exp\left(i\left(\vec{k}\cdot\vec{r} - \omega t \right) \right)
|
|
|
|
|
| 3948574226 |
\psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right)
|
|
|
|
|
| 3948574228 |
\psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
|
|
|
|
|
| 3948574230 |
\psi(\vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
|
|
|
|
|
| 3948574233 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
|
|
|
|
|
| 3948574235 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
|
|
|
|
|
| 3951205425 |
\vec{p}_{after} = \vec{p}_{1}
|
|
|
|
|
| 4147472132 |
E = \frac{h \omega}{2 \pi}
|
|
|
|
|
| 4298359835 |
E = \frac{1}{2}m v^2
|
|
|
|
|
| 4298359845 |
E = \frac{1}{2m}m^2 v^2
|
|
|
|
|
| 4298359851 |
E = \frac{p^2}{2m}
|
|
|
|
|
| 4341171256 |
i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{p^2}{2 m} \psi(\vec{r},t)
|
|
|
|
|
| 4348571256 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi(\vec{r},t)
|
|
|
|
|
| 4394958389 |
\vec{\nabla}\cdot \left(\vec{\nabla}\psi(\vec{r},t)\right) = \frac{i}{\hbar} \vec{\nabla}\cdot\left( \vec{p} \psi(\vec{r},t)\right)
|
|
|
|
|
| 4585828572 |
\epsilon_0 \mu_0 = \frac{1}{c^2}
|
|
|
|
|
| 4585932229 |
\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
|
|
|
|
|
| 4598294821 |
\exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
|
|
|
|
|
| 4638429483 |
\exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))
|
|
|
|
|
| 4742644828 |
\exp(i x)+\exp(-i x) = 2 \cos(x)
|
|
|
|
|
| 4827492911 |
\cos(2x)+(\sin(x))^2 = 1-(\sin(x))^2
|
|
|
|
|
| 4838429483 |
\exp(2 i x) = \cos(2 x)+i \sin(2 x)
|
|
|
|
|
| 4843995999 |
\frac{1}{2 i}\left(\exp(i x)-\exp(-i x)\right) = \sin(x)
|
|
|
|
|
| 4857472413 |
1 = \int \psi(x)\psi(x)^* dx
|
|
|
|
|
| 4857475848 |
\frac{1}{a^2} = \frac{W}{2}
|
|
|
|
|
| 4858429483 |
\exp(i x)\exp(i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))
|
|
|
|
|
| 4923339482 |
i x = \log(y)
|
|
|
|
|
| 4928239482 |
\log(y) = i x
|
|
|
|
|
| 4928923942 |
a = b
|
|
|
|
|
| 4929218492 |
a+b = c
|
|
|
|
|
| 4929482992 |
b = 2
|
|
|
|
|
| 4938429482 |
\exp(-i x) = \cos(x)+i \sin(-x)
|
|
|
|
|
| 4938429483 |
\exp(i x) = \cos(x)+i \sin(x)
|
|
|
|
|
| 4938429484 |
\exp(-i x) = \cos(x)-i \sin(x)
|
|
|
|
|
| 4943571230 |
\vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
|
|
|
|
|
| 4948934890 |
\langle \psi| \hat{A} |\psi\rangle = \langle a \rangle^*
|
|
|
|
|
| 4949359835 |
\langle x^2\rangle -2\langle x^2 \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 4954839242 |
\cos(2x)+i\sin(2x) = (\cos(x)+i \sin(x))(\cos(x)+i \sin(x))
|
|
|
|
|
| 4978429483 |
\exp(-i x) = \cos(-x)+i \sin(-x)
|
|
|
|
|
| 4984892984 |
a+2 = c
|
|
|
|
|
| 4985825552 |
\nabla^2 E(\vec{r})\exp(i \omega t) = i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 5530148480 |
\vec{p}_{1}-\vec{p}_{2} = \vec{p}_{electron}
|
|
|
|
|
| 5727578862 |
\frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)
|
|
|
|
|
| 5832984291 |
(\sin(x))^2 + (\cos(x))^2 = 1
|
|
|
|
|
| 5857434758 |
\int a dx = a x
|
|
|
|
|
| 5868688585 |
\frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = \frac{p^2}{2m} \psi(\vec{r},t)
|
|
|
|
|
| 5900595848 |
k = \frac{\omega}{v}
|
|
|
|
|
| 5928285821 |
x^2 + 2x(b/(2a)) + (b/(2a))^2 = (x+(b/(2a)))^2
|
|
|
|
|
| 5928292841 |
x^2 + (b/a)x + (b/(2a))^2 = -c/a + (b/(2a))^2
|
|
|
|
|
| 5938459282 |
x^2 + (b/a)x = -c/a
|
|
|
|
|
| 5958392859 |
x^2 + (b/a)x+(c/a) = 0
|
|
|
|
|
| 5959282914 |
x^2 + x(b/a) + (b/(2a))^2 = (x+(b/(2a)))^2
|
|
|
|
|
| 5982958248 |
x = -\sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
|
|
|
|
|
| 5982958249 |
x+(b/(2a)) = -\sqrt{(b/(2a))^2 - (c/a)}
|
|
|
|
|
| 5985371230 |
\vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi(\vec{r},t)
|
|
|
|
|
| 6742123016 |
\vec{p}_{electron}\cdot\vec{p}_{electron} = (\vec{p}_{1}\cdot\vec{p}_{1})+(\vec{p}_{2}\cdot\vec{p}_{2})-2(\vec{p}_{1}\cdot\vec{p}_{2})
|
|
|
|
|
| 7466829492 |
\vec{\nabla} \cdot \vec{E} = 0
|
|
|
|
|
| 7564894985 |
\int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)
|
|
|
|
|
| 7572664728 |
\cos(2x)+2(\sin(x))^2 = 1
|
|
|
|
|
| 7575738420 |
\left(\sin\left(\frac{n \pi}{W}x\right)\right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2}
|
|
|
|
|
| 7575859295 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859300 |
\epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{\nabla} \times \vec{E} )_k = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859302 |
\epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859304 |
\epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h}
|
|
|
|
|
| 7575859306 |
\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859308 |
\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right) = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859310 |
\hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859312 |
\vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7917051060 |
\vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2}
|
|
|
|
|
| 8139187332 |
\vec{p}_{1} = \vec{p}_{2}+\vec{p}_{electron}
|
|
|
|
|
| 8257621077 |
\vec{p}_{before} = \vec{p}_{1}
|
|
|
|
|
| 8311458118 |
\vec{p}_{after} = \vec{p}_{2}+\vec{p}_{electron}
|
|
|
|
|
| 8399484849 |
\langle x^2 -2x\langle x \rangle+\langle x \rangle^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 8484544728 |
-a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x)
|
|
|
|
|
| 8485757728 |
a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(kx)
|
|
|
|
|
| 8485867742 |
\frac{2}{W} = a^2
|
|
|
|
|
| 8489593958 |
d(u v) = u dv + v du
|
|
|
|
|
| 8489593960 |
d(u v) - v du = u dv
|
|
|
|
|
| 8489593962 |
u dv = d(u v) - v du
|
|
|
|
|
| 8489593964 |
\int u dv = u v - \int v du
|
|
|
|
|
| 8494839423 |
\nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
|
|
|
|
|
| 8572657110 |
1 = \int |\psi(x)|^2 dx
|
|
|
|
|
| 8572852424 |
\vec{E} = E(\vec{r},t)
|
|
|
|
|
| 8575746378 |
\int \frac{1}{2} dx = \frac{1}{2} x
|
|
|
|
|
| 8575748999 |
\frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x)\right) = -k^2 \left(a \sin(kx) + b \cos(kx)\right)
|
|
|
|
|
| 8576785890 |
1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
|
|
|
|
|
| 8577275751 |
0 = a \sin(0) + b\cos(0)
|
|
|
|
|
| 8582885111 |
\psi(x) = a \sin(kx) + b \cos(kx)
|
|
|
|
|
| 8582954722 |
x^2 + 2xh + h^2 = (x+h)^2
|
|
|
|
|
| 8849289982 |
\psi(x)^* = a \sin(\frac{n \pi}{W} x)
|
|
|
|
|
| 8889444440 |
1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right)\right)^2 dx
|
|
|
|
|
| 9059289981 |
\psi(x) = a \sin(k x)
|
|
|
|
|
| 9285928292 |
ax^2 + bx + c = 0
|
|
|
|
|
| 9291999979 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0\vec{\nabla} \times \frac{\partial \vec{H}}{\partial t}
|
|
|
|
|
| 9294858532 |
\hat{A}^+ = \hat{A}
|
|
|
|
|
| 9385938295 |
(x+(b/(2a)))^2 = -(c/a) + (b/(2a))^2
|
|
|
|
|
| 9393939991 |
\psi(x) = -\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
|
|
|
|
|
| 9393939992 |
\psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
|
|
|
|
|
| 9394939493 |
\nabla^2 E(\vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r},t)
|
|
|
|
|
| 9429829482 |
\frac{d}{dx} y = -\sin(x) + i\cos(x)
|
|
|
|
|
| 9482113948 |
\frac{dy}{y} = i dx
|
|
|
|
|
| 9482438243 |
(\cos(x))^2 = \cos(2x)+(\sin(x))^2
|
|
|
|
|
| 9482923849 |
\exp(i x) = y
|
|
|
|
|
| 9482928242 |
\cos(2x) = (\cos(x))^2 - (\sin(x))^2
|
|
|
|
|
| 9482928243 |
\cos(2x)+(\sin(x))^2 = (\cos(x))^2
|
|
|
|
|
| 9482943948 |
\log(y) = i dx
|
|
|
|
|
| 9482948292 |
b = c
|
|
|
|
|
| 9482984922 |
\frac{d}{dx} y = (i\sin(x) + \cos(x)) i
|
|
|
|
|
| 9483928192 |
\cos(2x)+i\sin(2x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
|
|
|
|
|
| 9485384858 |
\nabla^2 E(\vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 9485747245 |
\sqrt{\frac{2}{W}} = a
|
|
|
|
|
| 9485747246 |
-\sqrt{\frac{2}{W}} = a
|
|
|
|
|
| 9492920340 |
y = \cos(x)+i \sin(x)
|
|
|
|
|
| 9494829190 |
k
|
|
|
|
|
| 9495857278 |
\psi(x=W) = 0
|
|
|
|
|
| 9499428242 |
E(\vec{r},t) = E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 9499959299 |
a + k = b + k
|
|
|
|
|
| 9582958293 |
x = \sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
|
|
|
|
|
| 9582958294 |
x+(b/(2a)) = \sqrt{(b/(2a))^2 - (c/a)}
|
|
|
|
|
| 9584821911 |
c + d = a
|
|
|
|
|
| 9585727710 |
\psi(x=0) = 0
|
|
|
|
|
| 9596004948 |
x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle
|
|
|
|
|
| 9848292229 |
dy = y i dx
|
|
|
|
|
| 9848294829 |
\frac{d}{dx} y = y i
|
|
|
|
|
| 9858028950 |
\frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
|
|
|
|
|
| 9889984281 |
2(\sin(x))^2 = 1-\cos(2x)
|
|
|
|
|
| 9919999981 |
\rho = 0
|
|
|
|
|
| 9958485859 |
\frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
|
|
|
|
|
| 9988949211 |
(\sin(x))^2 = \frac{1-\cos(2x)}{2}
|
|
|
|
|
| 9991999979 |
\vec{\nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t}
|
|
|
|
|
| 9999998870 |
\frac{\vec{p}}{\hbar} = \vec{k}
|
|
|
|
|
| 9999999870 |
\frac{p}{\hbar} = k
|
|
|
|
|
| 9999999950 |
\beta = 1/(k_{Boltzmann}T)
|
|
|
|
|
| 9999999951 |
\langle x | k \rangle = \frac{\exp(i k x)}{\sqrt{2\pi}}
|
|
|
|
|
| 9999999952 |
f(x) \delta(x-a) = f(a)
|
|
|
|
|
| 9999999953 |
\int_{-\infty}^{\infty} \delta(x) dx = 1
|
|
|
|
|
| 9999999954 |
c = 1/\sqrt{\epsilon_0 \mu_0}
|
|
|
|
|
| 9999999955 |
\vec{E} = -\vec{\nabla} \Psi
|
|
|
|
|
| 9999999956 |
\vec{F} = \frac{\partial}{\partial t}\vec{p}
|
|
|
|
|
| 9999999957 |
\vec{F} = -\vec{\nabla} V
|
|
|
|
|
| 9999999960 |
\hbar = h/(2 \pi)
|
|
|
|
|
| 9999999961 |
\frac{E}{\hbar} = \omega
|
|
|
|
|
| 9999999962 |
p = \hbar k
|
|
|
|
|
| 9999999963 |
\lambda = h/p
|
|
|
|
|
| 9999999964 |
\omega = c k
|
|
|
|
|
| 9999999965 |
E = \omega \hbar
|
|
|
|
|
| 9999999966 |
\vec{L} = \vec{r}\times\vec{p}
|
|
|
|
|
| 9999999967 |
\vec{S} = \frac{1}{\mu_0} \vec{E}\times \vec{B}
|
|
|
|
|
| 9999999968 |
x = \frac{-b-\sqrt{b^2-4ac}}{2a}
|
|
|
|
|
| 9999999969 |
x = \frac{-b+\sqrt{b^2-4ac}}{2a}
|
|
|
|
|
| 9999999970 |
\eta_1 \sin\theta_1 = \eta_2 \sin\theta_2
|
|
|
|
|
| 9999999971 |
{\cal H} = \frac{p^2}{2m}+V
|
|
|
|
|
| 9999999972 |
{\cal H} |\psi\rangle = E |\psi\rangle
|
|
|
|
|
| 9999999973 |
\left( \Delta A \right)^2 = \langle A^2 \rangle - \langle A \rangle^2
|
|
|
|
|
| 9999999974 |
\langle \psi| \hat{A} |\psi\rangle = \int_{-\infty}^{\infty} \psi^* A \psi dx
|
|
|
|
|
| 9999999975 |
\langle \psi| \hat{A} |\psi\rangle = \langle a \rangle
|
|
|
|
|
| 9999999976 |
\hat{p} = -i \hbar \frac{\partial }{\partial x}
|
|
|
|
|
| 9999999977 |
[\hat{x},\hat{p}] = i \hbar
|
|
|
|
|
| 9999999978 |
\vec{\nabla} \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial E}{\partial t}
|
|
|
|
|
| 9999999979 |
\vec{\nabla} \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}
|
|
|
|
|
| 9999999980 |
\vec{\nabla} \cdot \vec{B} = 0
|
|
|
|
|
| 9999999981 |
\vec{\nabla} \cdot \vec{E} = \rho/\epsilon_0
|
|
|
|
|
| 9999999982 |
V = I R + Q/C + L \frac{\partial I}{\partial t}
|
|
|
|
|
| 9999999983 |
C = V A/d
|
|
|
|
|
| 9999999984 |
Q = C V
|
|
|
|
|
| 9999999985 |
V = I R
|
|
|
|
|
| 9999999986 |
\left[\right]\left[\right] = \left[\right]
|
|
|
|
|
| 9999999987 |
1 atmosphere = 101.325 Pascal
|
|
|
|
|
| 9999999988 |
1 atmosphere = 14.7 pounds/(inch^2)
|
|
|
|
|
| 9999999989 |
mass_{electron} = 511000 electronVolts/(q^2)
|
|
|
|
|
| 9999999990 |
Tesla = 10000*Gauss
|
|
|
|
|
| 9999999991 |
Pascal = Newton / (meter^2)
|
|
|
|
|
| 9999999992 |
Tesla = Newton / (Ampere*meter)
|
|
|
|
|
| 9999999993 |
Farad = Coulumb / Volt
|
|
|
|
|
| 9999999995 |
Volt = Joule / Coulumb
|
|
|
|
|
| 9999999996 |
Coulomb = Ampere / second
|
|
|
|
|
| 9999999997 |
Watt = Joule / second
|
|
|
|
|
| 9999999998 |
Joule = Newton*meter
|
|
|
|
|
| 9999999999 |
Newton = kilogram*meter/(second^2)
|
|
|
|
|
| 4961662865 |
x
|
|
|
|
|
| 9110536742 |
2 x
|
|
|
|
|
| 8483686863 |
\sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x)\right)
|
|
|
|
|
| 3470587782 |
\sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
|
|
|
|
|
| 8642992037 |
2
|
|
|
|
|
| 9894826550 |
2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \left(\exp(i x)+\exp(-i x)\right)
|
|
|
|
|
| 4257214632 |
\frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)
|
|
|
|
|
| 8699789241 |
2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)
|
|
|
|
|
| 9180861128 |
2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)
|
|
|
|
|
| 2405307372 |
\sin(2 x) = 2 \sin(x) \cos(x)
|
|
|
|
|
| 3268645065 |
x
|
|
|
|
|
| 9350663581 |
\pi
|
|
|
|
|
| 8332931442 |
\exp(i \pi) = \cos(\pi)+i \sin(\pi)
|
|
|
|
|
| 6885625907 |
\exp(i \pi) = -1 + i 0
|
|
|
|
|
| 3331824625 |
\exp(i \pi) = -1
|
|
|
|
|
| 4901237716 |
1
|
|
|
|
|
| 2501591100 |
\exp(i \pi) + 1 = 0
|
|
|
|
|
| 5136652623 |
E = KE + PE
|
|
mechanical energy is the sum of the potential plus kinetic energies |
|
|
| 7915321076 |
E, KE, PE
|
|
|
|
|
| 3192374582 |
E_2, KE_2, PE_2
|
|
|
|
|
| 7875206161 |
E_2 = KE_2 + PE_2
|
|
|
|
|
| 4229092186 |
E, KE, PE
|
|
|
|
|
| 1490821948 |
E_1, KE_1, PE_1
|
|
|
|
|
| 4303372136 |
E_1 = KE_1 + PE_1
|
|
|
|
|
| 5514556106 |
E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1)
|
|
|
|
|
| 8357234146 |
KE = \frac{1}{2} m v^2
|
|
kinetic energy |
Kinetic_energy
|
|
| 3658066792 |
KE_2, v_2
|
|
|
|
|
| 8082557839 |
KE, v
|
|
|
|
|
| 7676652285 |
KE_2 = \frac{1}{2} m v_2^2
|
|
|
|
|
| 2790304597 |
KE_1, v_1
|
|
|
|
|
| 9880327189 |
KE, v
|
|
|
|
|
| 4928007622 |
KE_1 = \frac{1}{2} m v_1^2
|
|
|
|
|
| 5733146966 |
KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right)
|
|
|
|
|
| 6554292307 |
t
|
|
|
|
|
| 4270680309 |
\frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t}
|
|
|
|
|
| 5781981178 |
x^2 - y^2 = (x+y)(x-y)
|
|
difference of squares |
Difference_of_two_squares
|
|
| 5946759357 |
v_2, v_1
|
|
|
|
|
| 6675701064 |
x, y
|
|
|
|
|
| 4648451961 |
v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1)
|
|
|
|
|
| 9356924046 |
\frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t}
|
|
|
|
|
| 9397152918 |
v = \frac{v_1 + v_2}{2}
|
|
average velocity |
|
|
| 2857430695 |
a = \frac{v_2 - v_1}{t}
|
|
acceleration |
|
|
| 7735737409 |
\frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t}
|
|
|
|
|
| 4784793837 |
\frac{KE_2 - KE_1}{t} = m v a
|
|
|
|
|
| 5345738321 |
F = m a
|
|
Newton's second law of motion |
Newton%27s_laws_of_motion#Newton's_second_law
|
|
| 2186083170 |
\frac{KE_2 - KE_1}{t} = v F
|
|
|
|
|
| 6715248283 |
PE = -F x
|
|
potential energy |
Potential_energy
|
|
| 3852078149 |
PE_2, x_2
|
|
|
|
|
| 7388921188 |
PE, x
|
|
|
|
|
| 2431507955 |
PE_2 = -F x_2
|
|
|
|
|
| 3412641898 |
PE_1, x_1
|
|
|
|
|
| 9937169749 |
PE, x
|
|
|
|
|
| 4669290568 |
PE_1 = -F x_1
|
|
|
|
|
| 7734996511 |
PE_2 - PE_1 = -F ( x_2 - x_1 )
|
|
|
|
|
| 2016063530 |
t
|
|
|
|
|
| 7267155233 |
\frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right)
|
|
|
|
|
| 9337785146 |
v = \frac{x_2 - x_1}{t}
|
|
average velocity |
|
|
| 4872970974 |
\frac{PE_2 - PE_1}{t} = -F v
|
|
|
|
|
| 2081689540 |
t
|
|
|
|
|
| 2770069250 |
\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t}
|
|
|
|
|
| 3591237106 |
\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v
|
|
|
|
|
| 1772416655 |
\frac{E_2 - E_1}{t} = v F - F v
|
|
|
|
|
| 1809909100 |
\frac{E_2 - E_1}{t} = 0
|
|
|
|
|
| 5778176146 |
t
|
|
|
|
|
| 3806977900 |
E_2 - E_1 = 0
|
|
|
|
|
| 5960438249 |
E_1
|
|
|
|
|
| 8558338742 |
E_2 = E_1
|
|
|
|
|
| 8945218208 |
\theta_{Brewster} + \theta_{refracted} = 90^{\circ}
|
|
|
based on figure 34-27 on page 824 in 2001_HRW
|
|
| 9025853427 |
\theta_{Brewster}
|
|
|
|
|
| 1310571337 |
\theta_{refracted} = 90^{\circ} - \theta_{Brewster}
|
|
|
|
|
| 6450985774 |
n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 )
|
|
Law of Refraction |
eq 34-44 on page 819 in 2001_HRW
|
|
| 1779497048 |
\theta_{Brewster}, \theta_{refraction}
|
|
|
|
|
| 7322344455 |
\theta_1, \theta_2
|
|
|
|
|
| 2575937347 |
n_1 \sin( \theta_{Brewster} ) = n_2 \sin( \theta_{refraction} )
|
|
|
|
|
| 7696214507 |
n_1 \sin( \theta_{Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{Brewster} )
|
|
|
|
|
| 8588429722 |
\sin( 90^{\circ} - x ) = \cos( x )
|
|
|
|
|
| 7375348852 |
\theta_{Brewster}
|
|
|
|
|
| 1512581563 |
x
|
|
|
|
|
| 6831637424 |
\sin( 90^{\circ} - \theta_{Brewster} ) = \cos( \theta_{Brewster} )
|
|
|
|
|
| 3061811650 |
n_1 \sin( \theta_{Brewster} ) = n_2 \cos( \theta_{Brewster} )
|
|
|
|
|
| 7857757625 |
n_1
|
|
|
|
|
| 9756089533 |
\sin( \theta_{Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{Brewster} )
|
|
|
|
|
| 5632428182 |
\cos( \theta_{Brewster} )
|
|
|
|
|
| 2768857871 |
\frac{\sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )} = \frac{n_2}{n_1}
|
|
|
|
|
| 4968680693 |
\tan( x ) = \frac{ \sin( x )}{\cos( x )}
|
|
|
|
|
| 7321695558 |
\theta_{Brewster}
|
|
|
|
|
| 9906920183 |
x
|
|
|
|
|
| 4501377629 |
\tan( \theta_{Brewster} ) = \frac{ \sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )}
|
|
|
|
|
| 3417126140 |
\tan( \theta_{Brewster} ) = \frac{ n_2 }{ n_1 }
|
|
|
|
|
| 5453995431 |
\arctan{ x }
|
|
|
|
|
| 6023986360 |
x
|
|
|
|
|
| 8495187962 |
\theta_{Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }
|
|
|
|
|
| 9040079362 |
f
|
|
|
|
|
| 9565166889 |
T
|
|
|
|
|
| 5426308937 |
v = \frac{d}{t}
|
|
|
|
|
| 7476820482 |
C
|
|
|
|
|
| 1277713901 |
d
|
|
|
|
|
| 6946088325 |
v = \frac{C}{t}
|
|
|
|
|
| 6785303857 |
C = 2 \pi r
|
|
|
|
|
| 4816195267 |
C, r
|
|
|
|
|
| 2113447367 |
C_{\rm{Earth\ orbit}}, r_{\rm{Earth\ orbit}}
|
|
|
|
|
| 6348260313 |
C_{\rm{Earth\ orbit}} = 2 \pi r_{\rm{Earth\ orbit}}
|
|
|
|
|
| 3847777611 |
C, v, t
|
|
|
|
|
| 6406487188 |
C_{\rm{Earth\ orbit}}, v_{\rm{Earth\ orbit}}, t_{\rm{Earth\ orbit}}
|
|
|
|
|
| 3046191961 |
v_{\rm{Earth\ orbit}} = \frac{C_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
|
|
|
|
|
| 3080027960 |
v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
|
|
|
|
|
| 7175416299 |
t_{\rm{Earth\ orbit}} = 1 {\rm{year}}
|
|
|
|
|
| 3219318145 |
\frac{365 {\rm days}}{1 {\rm year}} \frac{24 {\rm hours}}{1 {\rm day}} \frac{60 {\rm minutes}}{1 {\rm hour}} \frac{60 {\rm seconds}}{1 {\rm minute}}
|
|
|
|
|
| 8721295221 |
t_{\rm{Earth\ orbit}} = 3.16 10^7 {\rm{seconds}}
|
|
|
|
|
| 4593428198 |
v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{3.16\ 10^7 {\rm seconds}}
|
|
|
|
|
| 3472836147 |
r_{\rm{Earth\ orbit}} = 1.496\ 10^8 {\rm km}
|
|
|
|
|
| 6998364753 |
v_{\rm{Earth\ orbit}} = \frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}}
|
|
|
|
|
| 4180845508 |
v_{\rm{Earth\ orbit}} = 29.8 \frac{{\rm km}}{{\rm sec}}
|
|
|
|
|
| 4662369843 |
x' = \gamma (x - v t)
|
|
|
|
|
| 2983053062 |
x = \gamma (x' + v t')
|
|
|
|
|
| 3426941928 |
x = \gamma ( \gamma (x - v t) + v t' )
|
|
|
|
|
| 2096918413 |
x = \gamma ( \gamma x - \gamma v t + v t' )
|
|
|
|
|
| 7741202861 |
x = \gamma^2 x - \gamma^2 v t + \gamma v t'
|
|
|
|
|
| 7337056406 |
\gamma^2 x
|
|
|
|
|
| 4139999399 |
x - \gamma^2 x = - \gamma^2 v t + \gamma v t'
|
|
|
|
|
| 3495403335 |
x
|
|
|
|
|
| 9031609275 |
x (1 - \gamma^2 ) = - \gamma^2 v t + \gamma v t'
|
|
|
|
|
| 8014566709 |
\gamma^2 v t
|
|
|
|
|
| 9409776983 |
x (1 - \gamma^2 ) + \gamma^2 v t = \gamma v t'
|
|
|
|
|
| 2226340358 |
\gamma v
|
|
|
|
|
| 6417705759 |
\frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = \gamma v t'
|
|
|
|
|
| 8730201316 |
\frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t = t'
|
|
|
first term was multiplied by \gamma/\gamma
|
|
| 1974334644 |
\frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = t'
|
|
|
|
|
| 5148266645 |
t' = \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t
|
|
|
|
|
| 4287102261 |
x^2 + y^2 + z^2 = c^2 t^2
|
|
|
describes a spherical wavefront
|
|
| 1201689765 |
x'^2 + y'^2 + z'^2 = c^2 t'^2
|
|
|
describes a spherical wavefront for an observer in a moving frame of reference
|
|
| 7057864873 |
y' = y
|
|
|
frame of reference is moving only along x direction
|
|
| 8515803375 |
z' = z
|
|
|
frame of reference is moving only along x direction
|
|
| 6153463771 |
3
|
|
|
|
|
| 4746576736 |
asdf
|
|
|
|
|
| 1027270623 |
minga
|
|
|
|
|
| 6101803533 |
ming
|
|
|
|
|
| 9231495607 |
a = b
|
|
|
|
|
| 8664264194 |
b + 2
|
|
|
|
|
| 9805063945 |
\gamma^2 (x - v t)^2 + y^2 + z^2 = c^2 \gamma^2 \left( t + \frac{ 1 - \gamma^2 }{ \gamma^2 } \frac{x}{v} \right)^2
|
|
|
|
|
| 4057686137 |
C \to C_{\rm{Earth\ orbit}}
|
|
|
|
|
| 2346150725 |
r \to r_{\rm{Earth\ orbit}}
|
|
|
|
|
| 9753878784 |
v \to v_{\rm{Earth\ orbit}}
|
|
|
|
|
| 8135396036 |
t \to t_{\rm{Earth\ orbit}}
|
|
|
|
|
| 2773628333 |
\theta_1 \to \theta_{Brewster}
|
|
|
|
|
| 7154592211 |
\theta_2 \to \theta_{\rm refracted}
|
|
|
|
|
| 3749492596 |
E \to E_1
|
|
|
|
|
| 1258245373 |
E \to E_2
|
|
|
|
|
| 4147101187 |
KE \to KE_1
|
|
|
|
|
| 3809726424 |
PE \to PE_1
|
|
|
|
|
| 6383056612 |
KE \to KE_2
|
|
|
|
|
| 5075406409 |
PE \to PE_2
|
|
|
|
|
| 5398681503 |
v \to v_1
|
|
|
|
|
| 9305761407 |
v \to v_2
|
|
|
|
|
| 4218009993 |
x \to x_1
|
|
|
|
|
| 4188639044 |
x \to x_2
|
|
|
|
|
| 8173074178 |
x \to v_2
|
|
|
|
|
| 1025759423 |
y \to v_1
|
|
|
|
|
| 1935543849 |
\gamma^2 x^2 - \gamma^2 2 x v t + \gamma^2 v^2 t^2 + y^2 + z^2 = c^2 \gamma^2 \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x^2}{\gamma^2} + c^2 \gamma^2 2 t \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x}{\gamma} + c^2 \gamma^2 t^2
|
|
|
|
|
| 1586866563 |
\left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right) = t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)
|
|
|
|
|
| 3182633789 |
\gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1
|
|
|
|
|
| 1916173354 |
-\gamma^2 v^2 + c^2 \gamma^2 = c^2
|
|
|
|
|
| 2076171250 |
-\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} = 0
|
|
|
|
|
| 5284610349 |
\gamma^2
|
|
|
|
|
| 2417941373 |
- c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 - \gamma^2
|
|
|
|
|
| 5787469164 |
1 - \gamma^2
|
|
|
|
|
| 1639827492 |
- c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2} = 1
|
|
|
|
|
| 5669500954 |
v^2 \gamma^2
|
|
|
|
|
| 5763749235 |
-c^2 + c^2 \gamma^2 = v^2 \gamma^2
|
|
|
|
|
| 6408214498 |
c^2
|
|
|
|
|
| 2999795755 |
c^2 \gamma^2 = v^2 \gamma^2 + c^2
|
|
|
|
|
| 3412946408 |
v^2 \gamma^2
|
|
|
|
|
| 2542420160 |
c^2 \gamma^2 - v^2 \gamma^2 = c^2
|
|
|
|
|
| 7743841045 |
\gamma^2
|
|
|
|
|
| 7513513483 |
\gamma^2 (c^2 - v^2) = c^2
|
|
|
|
|
| 8571466509 |
c^2 - \gamma^2
|
|
|
|
|
| 7906112355 |
\gamma^2 = \frac{c^2}{c^2 - \gamma^2}
|
|
|
|
|
| 3060139639 |
1/2
|
|
|
|
|
| 1528310784 |
\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
|
|
|
|
|
| 8360117126 |
\gamma = \frac{-1}{\sqrt{1-\frac{v^2}{c^2}}}
|
|
|
not a physically valid result in this context
|
|
| 3366703541 |
a = \frac{v - v_0}{t}
|
|
|
acceleration is the average change in speed over a duration
|
|
| 7083390553 |
t
|
|
|
|
|
| 4748157455 |
a t = v - v_0
|
|
|
|
|
| 6417359412 |
v_0
|
|
|
|
|
| 4798787814 |
a t + v_0 = v
|
|
|
|
|
| 3462972452 |
v = v_0 + a t
|
|
|
|
|
| 3411994811 |
v_{\rm ave} = \frac{d}{t}
|
|
|
|
|
| 6175547907 |
v_{\rm ave} = \frac{v + v_0}{2}
|
|
|
|
|
| 9897284307 |
\frac{d}{t} = \frac{v + v_0}{2}
|
|
|
|
|
| 8865085668 |
t
|
|
|
|
|
| 8706092970 |
d = \left(\frac{v + v_0}{2}\right)t
|
|
|
|
|
| 7011114072 |
d = \frac{(v_0 + a t) + v_0}{2} t
|
|
|
|
|
| 1265150401 |
d = \frac{2 v_0 + a t}{2} t
|
|
|
|
|
| 9658195023 |
d = v_0 t + \frac{1}{2} a t^2
|
|
|
|
|
| 5799753649 |
2
|
|
|
|
|
| 7215099603 |
v^2 = v_0^2 + 2 a t v_0 + a^2 t^2
|
|
|
|
|
| 5144263777 |
v^2 = v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right)
|
|
|
|
|
| 7939765107 |
v^2 = v_0^2 + 2 a d
|
|
|
|
|
| 6729698807 |
v_0
|
|
|
|
|
| 9759901995 |
v - v_0 = a t
|
|
|
|
|
| 2242144313 |
a
|
|
|
|
|
| 1967582749 |
t = \frac{v - v_0}{a}
|
|
|
|
|
| 5733721198 |
d = \frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right)
|
|
|
|
|
| 5611024898 |
d = \frac{1}{2 a} (v^2 - v_0^2)
|
|
|
|
|
| 5542390646 |
2 a
|
|
|
|
|
| 8269198922 |
2 a d = v^2 - v_0^2
|
|
|
|
|
| 9070454719 |
v_0^2
|
|
|
|
|
| 4948763856 |
2 a d + v_0^2 = v^2
|
|
|
|
|
| 9645178657 |
a t
|
|
|
|
|
| 6457044853 |
v - a t = v_0
|
|
|
|
|
| 1259826355 |
d = (v - a t) t + \frac{1}{2} a t^2
|
|
|
|
|
| 4580545876 |
d = v t - a t^2 + \frac{1}{2} a t^2
|
|
|
|
|
| 6421241247 |
d = v t - \frac{1}{2} a t^2
|
|
|
|
|
| 4182362050 |
Z = |Z| \exp( i \theta )
|
|
|
Z \in \mathbb{C}
|
|
| 1928085940 |
Z^* = |Z| \exp( -i \theta )
|
|
|
|
|
| 9191880568 |
Z Z^* = |Z| |Z| \exp( -i \theta ) \exp( i \theta )
|
|
|
|
|
| 3350830826 |
Z Z^* = |Z|^2
|
|
|
|
|
| 8396997949 |
I = | A + B |^2
|
|
|
intensity of two waves traveling opposite directions on same path
|
|
| 4437214608 |
Z
|
|
|
|
|
| 5623794884 |
A + B
|
|
|
|
|
| 2236639474 |
(A + B)(A + B)^* = |A + B|^2
|
|
|
|
|
| 1020854560 |
I = (A + B)(A + B)^*
|
|
|
|
|
| 6306552185 |
I = (A + B)(A^* + B^*)
|
|
|
|
|
| 8065128065 |
I = A A^* + B B^* + A B^* + B A^*
|
|
|
|
|
| 9761485403 |
Z
|
|
|
|
|
| 8710504862 |
A
|
|
|
|
|
| 4075539836 |
A A^* = |A|^2
|
|
|
|
|
| 6529120965 |
B
|
|
|
|
|
| 1511199318 |
Z
|
|
|
|
|
| 7107090465 |
B B^* = |B|^2
|
|
|
|
|
| 5125940051 |
I = |A|^2 + B B^* + A B^* + B A^*
|
|
|
|
|
| 1525861537 |
I = |A|^2 + |B|^2 + A B^* + B A^*
|
|
|
|
|
| 1742775076 |
Z \to B
|
|
|
|
|
| 7607271250 |
\theta \to \phi
|
|
|
|
|
| 4192519596 |
B = |B| \exp(i \phi)
|
|
|
|
|
| 2064205392 |
A
|
|
|
|
|
| 1894894315 |
Z
|
|
|
|
|
| 1357848476 |
A = |A| \exp(i \theta)
|
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|
|
|
| 6555185548 |
A^* = |A| \exp(-i \theta)
|
|
|
|
|
| 4504256452 |
B^* = |B| \exp(-i \phi)
|
|
|
|
|
| 7621705408 |
I = |A|^2 + |B|^2 + |A| |B| \exp(-i \theta) \exp(i \phi) + |A| |B| \exp(i \theta) \exp(-i \phi)
|
|
|
|
|
| 3085575328 |
I = |A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi))
|
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|
|
|
| 4935235303 |
x
|
|
|
|
|
| 2293352649 |
\theta - \phi
|
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| 3660957533 |
\cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
|
|
|
|
|
| 3967985562 |
2
|
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|
|
| 2700934933 |
2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
|
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|
|
|
| 8497631728 |
I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )
|
|
|
|
|
| 6774684564 |
\theta = \phi
|
|
|
for coherent waves
|
|
| 2099546947 |
I = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )
|
|
|
|
|
| 2719691582 |
|A| = |B|
|
|
|
in a loop
|
|
| 3257276368 |
I = |A|^2 + |A|^2 + |A| |A| 2
|
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|
|
|
| 8283354808 |
I_{\rm coherent} = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )
|
|
|
|
|
| 8046208134 |
I_{\rm coherent} = |A|^2 + |A|^2 + |A| |A| 2
|
|
|
|
|
| 1172039918 |
I_{\rm coherent} = 4 |A|^2
|
|
|
|
|
| 8602221482 |
\langle \cos(\theta - \phi) \rangle = 0
|
|
|
incoherent light source
|
|
| 6240206408 |
I_{\rm incoherent} = |A|^2 + |B|^2
|
|
|
|
|
| 6529793063 |
I_{\rm incoherent} = |A|^2 + |A|^2
|
|
|
|
|
| 3060393541 |
I_{\rm incoherent} = 2|A|^2
|
|
|
|
|
| 6556875579 |
\frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2
|
|
|
|
|
| 8750500372 |
f
|
|
|
|
|
| 7543102525 |
g
|
|
|
|
|
| 8267133829 |
a = b
|
|
|
|
|
| 7968822469 |
c = d
|
|
|
|
|
| 3169580383 |
\vec{a} = \frac{d\vec{v}}{dt}
|
|
|
acceleration is the change in speed over a duration
|
|
| 8602512487 |
\vec{a} = a_x \hat{x} + a_y \hat{y}
|
|
|
decompose acceleration into two components
|
|
| 4158986868 |
a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt}
|
|
|
|
|
| 5349866551 |
\vec{v} = v_x \hat{x} + v_y \hat{y}
|
|
|
|
|
| 7729413831 |
a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right)
|
|
|
|
|
| 1819663717 |
a_x = \frac{d}{dt} v_x
|
|
|
|
|
| 8228733125 |
a_y = \frac{d}{dt} v_y
|
|
|
|
|
| 9707028061 |
a_x = 0
|
|
|
|
|
| 2741489181 |
a_y = -g
|
|
|
|
|
| 3270039798 |
2
|
|
|
|
|
| 8880467139 |
2
|
|
|
|
|
| 8750379055 |
0 = \frac{d}{dt} v_x
|
|
|
|
|
| 1977955751 |
-g = \frac{d}{dt} v_y
|
|
|
|
|
| 6672141531 |
dt
|
|
|
|
|
| 1702349646 |
-g dt = d v_y
|
|
|
|
|
| 8584698994 |
-g \int dt = \int d v_y
|
|
|
|
|
| 9973952056 |
-g t = v_y - v_{0, y}
|
|
|
|
|
| 4167526462 |
v_{0, y}
|
|
|
|
|
| 6572039835 |
-g t + v_{0, y} = v_y
|
|
|
|
|
| 7252338326 |
v_y = \frac{dy}{dt}
|
|
|
|
|
| 6204539227 |
-g t + v_{0, y} = \frac{dy}{dt}
|
|
|
|
|
| 1614343171 |
dt
|
|
|
|
|
| 8145337879 |
-g t dt + v_{0, y} dt = dy
|
|
|
|
|
| 8808860551 |
-g \int t dt + v_{0, y} \int dt = \int dy
|
|
|
|
|
| 2858549874 |
- \frac{1}{2} g t^2 + v_{0, y} t = y - y_0
|
|
|
|
|
| 6098638221 |
y_0
|
|
|
|
|
| 2461349007 |
- \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y
|
|
|
|
|
| 8717193282 |
dt
|
|
|
|
|
| 1166310428 |
0 dt = d v_x
|
|
|
|
|
| 2366691988 |
\int 0 dt = \int d v_x
|
|
|
|
|
| 1676472948 |
0 = v_x - v_{0, x}
|
|
|
|
|
| 1439089569 |
v_{0, x}
|
|
|
|
|
| 6134836751 |
v_{0, x} = v_x
|
|
|
|
|
| 8460820419 |
v_x = \frac{dx}{dt}
|
|
|
|
|
| 7455581657 |
v_{0, x} = \frac{dx}{dt}
|
|
|
|
|
| 8607458157 |
dt
|
|
|
|
|
| 1963253044 |
v_{0, x} dt = dx
|
|
|
|
|
| 3676159007 |
v_{0, x} \int dt = \int dx
|
|
|
|
|
| 9882526611 |
v_{0, x} t = x - x_0
|
|
|
|
|
| 3182907803 |
x_0
|
|
|
|
|
| 8486706976 |
v_{0, x} t + x_0 = x
|
|
|
|
|
| 1306360899 |
x = v_{0, x} t + x_0
|
|
|
|
|
| 7049769409 |
2
|
|
|
|
|
| 9341391925 |
\vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y}
|
|
|
|
|
| 6410818363 |
\theta
|
|
|
|
|
| 7391837535 |
\cos(\theta) = \frac{v_{0, x}}{v_0}
|
|
|
|
|
| 7376526845 |
\sin(\theta) = \frac{v_{0, y}}{v_0}
|
|
|
|
|
| 5868731041 |
v_0
|
|
|
|
|
| 6083821265 |
v_0 \cos(\theta) = v_{0, x}
|
|
|
|
|
| 5438722682 |
x = v_0 t \cos(\theta) + x_0
|
|
|
|
|
| 5620558729 |
v_0
|
|
|
|
|
| 8949329361 |
v_0 \sin(\theta) = v_{0, y}
|
|
|
|
|
| 1405465835 |
y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0
|
|
|
|
|
| 9862900242 |
y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0
|
|
|
|
|
| 6050070428 |
v_{0, x}
|
|
|
|
|
| 3274926090 |
t = \frac{x - x_0}{v_{0, x}}
|
|
|
|
|
| 7354529102 |
y = - \frac{1}{2} g \left( \frac{x - x_0}{v_{0, x}} \right)^2 + v_{0, y} \frac{x - x_0}{v_{0, x}} + y_0
|
|
|
|
|
| 8406170337 |
y \to y_f
|
|
|
|
|
| 2403773761 |
t \to t_f
|
|
|
|
|
| 5379546684 |
y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
|
|
|
|
|
| 5373931751 |
t = t_f
|
|
|
|
|
| 9112191201 |
y_f = 0
|
|
|
|
|
| 8198310977 |
0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
|
|
|
|
|
| 1650441634 |
y_0 = 0
|
|
|
define coordinate system such that initial height is at origin
|
|
| 1087417579 |
0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)
|
|
|
|
|
| 4829590294 |
t_f
|
|
|
|
|
| 2086924031 |
0 = - \frac{1}{2} g t_f + v_0 \sin(\theta)
|
|
|
|
|
| 6974054946 |
\frac{1}{2} g t_f
|
|
|
|
|
| 1191796961 |
\frac{1}{2} g t_f = v_0 \sin(\theta)
|
|
|
|
|
| 2510804451 |
2/g
|
|
|
|
|
| 4778077984 |
t_f = \frac{2 v_0 \sin(\theta)}{g}
|
|
|
|
|
| 3273630811 |
x \to x_f
|
|
|
|
|
| 6732786762 |
t \to t_f
|
|
|
|
|
| 3485125659 |
x_f = v_0 t_f \cos(\theta) + x_0
|
|
|
|
|
| 4370074654 |
t = t_f
|
|
|
|
|
| 2378095808 |
x_f = x_0 + d
|
|
|
|
|
| 4268085801 |
x_0 + d = v_0 t_f \cos(\theta) + x_0
|
|
|
|
|
| 8072682558 |
x_0
|
|
|
|
|
| 7233558441 |
d = v_0 t_f \cos(\theta)
|
|
|
|
|
| 2297105551 |
d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)
|
|
|
|
|
| 7587034465 |
\theta
|
|
|
|
|
| 7214442790 |
x
|
|
|
|
|
| 2519058903 |
\sin(2 \theta) = 2 \sin(\theta) \cos(\theta)
|
|
|
|
|
| 8922441655 |
d = \frac{v_0^2}{g} \sin(2 \theta)
|
|
|
|
|
| 5667870149 |
\theta
|
|
|
|
|
| 1541916015 |
\theta = \frac{\pi}{4}
|
|
|
|
|
| 3607070319 |
d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)
|
|
|
|
|
| 5353282496 |
d = \frac{v_0^2}{g}
|
|
|
|
|
| 0000040490 |
a^2
|
|
|
|
|
| 0000999900 |
b/(2a)
|
|
|
|
|
| 0001030901 |
\cos(x)
|
|
|
|
|
| 0001111111 |
(\sin(x))^2
|
|
|
|
|
| 0001209482 |
2 \pi
|
|
|
|
|
| 0001304952 |
\hbar
|
|
|
|
|
| 0001334112 |
W
|
|
|
|
|
| 0001921933 |
2 i
|
|
|
|
|
| 0002239424 |
2
|
|
|
|
|
| 0002338514 |
\vec{p}_{2}
|
|
|
|
|
| 0002342425 |
m/m
|
|
|
|
|
| 0002367209 |
1/2
|
|
|
|
|
| 0002393922 |
x
|
|
|
|
|
| 0002424922 |
a
|
|
|
|
|
| 0002436656 |
i \hbar
|
|
|
|
|
| 0002449291 |
b/(2a)
|
|
|
|
|
| 0002838490 |
b/(2a)
|
|
|
|
|
| 0002919191 |
\sin(-x)
|
|
|
|
|
| 0002929944 |
1/2
|
|
|
|
|
| 0002940021 |
2 \pi
|
|
|
|
|
| 0003232242 |
t
|
|
|
|
|
| 0003413423 |
\cos(-x)
|
|
|
|
|
| 0003747849 |
-1
|
|
|
|
|
| 0003838111 |
2
|
|
|
|
|
| 0003919391 |
x
|
|
|
|
|
| 0003949052 |
-x
|
|
|
|
|
| 0003949921 |
\hbar
|
|
|
|
|
| 0003954314 |
dx
|
|
|
|
|
| 0003981813 |
-\sin(x)
|
|
|
|
|
| 0004089571 |
2x
|
|
|
|
|
| 0004264724 |
y
|
|
|
|
|
| 0004307451 |
(b/(2a))^2
|
|
|
|
|
| 0004582412 |
x
|
|
|
|
|
| 0004829194 |
2
|
|
|
|
|
| 0004831494 |
a
|
|
|
|
|
| 0004849392 |
x
|
|
|
|
|
| 0004858592 |
h
|
|
|
|
|
| 0004934845 |
x
|
|
|
|
|
| 0004948585 |
a
|
|
|
|
|
| 0005395034 |
a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
|
|
|
|
|
| 0005626421 |
t
|
|
|
|
|
| 0005749291 |
f
|
|
|
|
|
| 0005938585 |
\frac{-\hbar^2}{2m}
|
|
|
|
|
| 0006466214 |
(\sin(x))^2
|
|
|
|
|
| 0006544644 |
t
|
|
|
|
|
| 0006563727 |
x
|
|
|
|
|
| 0006644853 |
c/a
|
|
|
|
|
| 0006656532 |
e
|
|
|
|
|
| 0007471778 |
2(\sin(x))^2
|
|
|
|
|
| 0007563791 |
i
|
|
|
|
|
| 0007636749 |
x
|
|
|
|
|
| 0007894942 |
(\sin(x))^2
|
|
|
|
|
| 0008837284 |
T
|
|
|
|
|
| 0008842811 |
\cos(2x)
|
|
|
|
|
| 0009458842 |
\psi(x)
|
|
|
|
|
| 0009484724 |
\frac{n \pi}{W}x
|
|
|
|
|
| 0009485857 |
a^2\frac{2}{W}
|
|
|
|
|
| 0009485858 |
\frac{2n\pi}{W}
|
|
|
|
|
| 0009492929 |
v du
|
|
|
|
|
| 0009587738 |
\psi
|
|
|
|
|
| 0009594995 |
1/2
|
|
|
|
|
| 0009877781 |
y
|
|
|
|
|
| 0203024440 |
1 = \int_0^W a \sin(\frac{n \pi}{W} x) \psi(x)^* dx
|
|
|
|
|
| 0404050504 |
\lambda = \frac{v}{f}
|
|
|
|
|
| 0439492440 |
\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)\right|_0^W
|
|
|
|
|
| 0934990943 |
k = \frac{2 \pi}{v T}
|
|
|
|
|
| 0948572140 |
\int \cos(a x) dx = \frac{1}{a}\sin(a x)
|
|
|
|
|
| 1010393913 |
\langle \psi| \hat{A}^+ |\psi \rangle = \langle a \rangle^*
|
|
|
|
|
| 1010393944 |
x = \langle\psi_{\alpha}| a_{\beta} |\psi_{\beta} \rangle
|
|
|
|
|
| 1010923823 |
k W = n \pi
|
|
|
|
|
| 1020010291 |
0 = a \sin(k W)
|
|
|
|
|
| 1020394900 |
p = h/\lambda
|
|
|
|
|
| 1020394902 |
E = h f
|
|
|
|
|
| 1029039903 |
p = m v
|
|
|
|
|
| 1029039904 |
p^2 = m^2 v^2
|
|
|
|
|
| 1158485859 |
\frac{-\hbar^2}{2m} \nabla^2 = {\cal H}
|
|
|
|
|
| 1202310110 |
\frac{1}{a^2} = \int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
|
|
|
|
|
| 1202312210 |
\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
|
|
|
|
|
| 1203938249 |
a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle = a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
|
|
|
|
|
| 1248277773 |
\cos(2x) = 1-2(\sin(x))^2
|
|
|
|
|
| 1293913110 |
0 = b
|
|
|
|
|
| 1293923844 |
\lambda = v T
|
|
|
|
|
| 1314464131 |
\vec{\nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
|
|
|
|
|
| 1314864131 |
\vec{\nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E}
|
|
|
|
|
| 1395858355 |
x = \langle \psi_{\alpha}| a_{\alpha} |\psi_{\beta}\rangle
|
|
|
|
|
| 1492811142 |
f = x - d
|
|
|
|
|
| 1492842000 |
\nabla \vec{x} = f(y)
|
|
|
|
|
| 1636453295 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = - \nabla^2 \vec{E}
|
|
|
|
|
| 1638282134 |
\vec{p}_{before} = \vec{p}_{after}
|
|
|
|
|
| 1648958381 |
\nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{\nabla}\psi(\vec{r},t)\right)
|
|
|
|
|
| 1857710291 |
0 = a \sin(n \pi)
|
|
|
|
|
| 1858578388 |
\nabla^2 E(\vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 1858772113 |
k = \frac{n \pi}{W}
|
|
|
|
|
| 1934748140 |
\int |\psi(x)|^2 dx = 1
|
|
|
|
|
| 2029293929 |
\nabla^2 E(\vec{r})\exp(i \omega t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 2103023049 |
\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right)
|
|
|
|
|
| 2113211456 |
f = 1/T
|
|
|
|
|
| 2123139121 |
-\exp(-i x) = -\cos(x)+i \sin(x)
|
|
|
|
|
| 2131616531 |
T f = 1
|
|
|
|
|
| 2258485859 |
{\cal H} \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
|
|
|
|
|
| 2394240499 |
x = a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle
|
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|
|
|
| 2394853829 |
\exp(-i x) = \cos(-x)+i \sin(-x)
|
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|
| 2394935831 |
( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
|
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|
|
|
| 2394935835 |
\left(\langle\psi| \hat{A} |\psi\rangle \right)^+ = \left(\langle a \rangle\right)^+
|
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|
| 2395958385 |
\nabla^2 \psi\left(\vec{r},t)\right) = \frac{-p^2}{\hbar} \psi(\vec{r},t)
|
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|
|
|
| 2404934990 |
\langle x^2\rangle -2\langle x \rangle\langle x \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
|
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|
| 2494533900 |
\langle x^2\rangle -\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
|
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|
| 2569154141 |
\vec{\nabla} \times \frac{\partial}{\partial t}\vec{H} = \epsilon_0 \frac{\partial^2 }{\partial t^2}\vec{E}
|
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|
|
|
| 2648958382 |
\nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p}\psi(\vec{r},t)\right)
|
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|
| 2848934890 |
\langle a \rangle^* = \langle a \rangle
|
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|
|
|
| 2944838499 |
\psi(x) = a \sin(\frac{n \pi}{W} x)
|
|
|
|
|
| 3121234211 |
\frac{k}{2\pi} = \lambda
|
|
|
|
|
| 3121234212 |
p = \frac{h k}{2\pi}
|
|
|
|
|
| 3121513111 |
k = \frac{2 \pi}{\lambda}
|
|
|
|
|
| 3131111133 |
T = 1 / f
|
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|
|
|
| 3131211131 |
\omega = 2 \pi f
|
|
|
|
|
| 3132131132 |
\omega = \frac{2\pi}{T}
|
|
|
|
|
| 3147472131 |
\frac{\omega}{2 \pi} = f
|
|
|
|
|
| 3285732911 |
(\cos(x))^2 = 1-(\sin(x))^2
|
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|
|
|
| 3485475729 |
\nabla^2 E(\vec{r}) = - \frac{\omega^2}{c^2} E(\vec{r})
|
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|
| 3585845894 |
\langle \left(x-\langle x \rangle\right)^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
|
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|
| 3829492824 |
\frac{1}{2}\left(\exp(i x)+\exp(-i x)\right) = \cos(x)
|
|
|
|
|
| 3924948349 |
a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
|
|
|
|
|
| 3942849294 |
\exp(i x)-\exp(-i x) = 2 i \sin(x)
|
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|
|
|
| 3943939590 |
x = a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle
|
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|
|
| 3947269979 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
|
|
|
|
|
| 3948571256 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi(\vec{r},t)
|
|
|
|
|
| 3948572230 |
\vec{\nabla}\psi(\vec{r},t) = \psi_0 \vec{\nabla}\exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
|
|
|
|
|
| 3948574224 |
\psi(\vec{r},t) = \psi_0 \exp\left(i\left(\vec{k}\cdot\vec{r} - \omega t \right) \right)
|
|
|
|
|
| 3948574226 |
\psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right)
|
|
|
|
|
| 3948574228 |
\psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
|
|
|
|
|
| 3948574230 |
\psi(\vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
|
|
|
|
|
| 3948574233 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
|
|
|
|
|
| 3948574235 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
|
|
|
|
|
| 3951205425 |
\vec{p}_{after} = \vec{p}_{1}
|
|
|
|
|
| 4147472132 |
E = \frac{h \omega}{2 \pi}
|
|
|
|
|
| 4298359835 |
E = \frac{1}{2}m v^2
|
|
|
|
|
| 4298359845 |
E = \frac{1}{2m}m^2 v^2
|
|
|
|
|
| 4298359851 |
E = \frac{p^2}{2m}
|
|
|
|
|
| 4341171256 |
i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{p^2}{2 m} \psi(\vec{r},t)
|
|
|
|
|
| 4348571256 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi(\vec{r},t)
|
|
|
|
|
| 4394958389 |
\vec{\nabla}\cdot \left(\vec{\nabla}\psi(\vec{r},t)\right) = \frac{i}{\hbar} \vec{\nabla}\cdot\left( \vec{p} \psi(\vec{r},t)\right)
|
|
|
|
|
| 4585828572 |
\epsilon_0 \mu_0 = \frac{1}{c^2}
|
|
|
|
|
| 4585932229 |
\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
|
|
|
|
|
| 4598294821 |
\exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
|
|
|
|
|
| 4638429483 |
\exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))
|
|
|
|
|
| 4742644828 |
\exp(i x)+\exp(-i x) = 2 \cos(x)
|
|
|
|
|
| 4827492911 |
\cos(2x)+(\sin(x))^2 = 1-(\sin(x))^2
|
|
|
|
|
| 4838429483 |
\exp(2 i x) = \cos(2 x)+i \sin(2 x)
|
|
|
|
|
| 4843995999 |
\frac{1}{2 i}\left(\exp(i x)-\exp(-i x)\right) = \sin(x)
|
|
|
|
|
| 4857472413 |
1 = \int \psi(x)\psi(x)^* dx
|
|
|
|
|
| 4857475848 |
\frac{1}{a^2} = \frac{W}{2}
|
|
|
|
|
| 4858429483 |
\exp(i x)\exp(i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))
|
|
|
|
|
| 4923339482 |
i x = \log(y)
|
|
|
|
|
| 4928239482 |
\log(y) = i x
|
|
|
|
|
| 4928923942 |
a = b
|
|
|
|
|
| 4929218492 |
a+b = c
|
|
|
|
|
| 4929482992 |
b = 2
|
|
|
|
|
| 4938429482 |
\exp(-i x) = \cos(x)+i \sin(-x)
|
|
|
|
|
| 4938429483 |
\exp(i x) = \cos(x)+i \sin(x)
|
|
|
|
|
| 4938429484 |
\exp(-i x) = \cos(x)-i \sin(x)
|
|
|
|
|
| 4943571230 |
\vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
|
|
|
|
|
| 4948934890 |
\langle \psi| \hat{A} |\psi\rangle = \langle a \rangle^*
|
|
|
|
|
| 4949359835 |
\langle x^2\rangle -2\langle x^2 \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 4954839242 |
\cos(2x)+i\sin(2x) = (\cos(x)+i \sin(x))(\cos(x)+i \sin(x))
|
|
|
|
|
| 4978429483 |
\exp(-i x) = \cos(-x)+i \sin(-x)
|
|
|
|
|
| 4984892984 |
a+2 = c
|
|
|
|
|
| 4985825552 |
\nabla^2 E(\vec{r})\exp(i \omega t) = i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 5530148480 |
\vec{p}_{1}-\vec{p}_{2} = \vec{p}_{electron}
|
|
|
|
|
| 5727578862 |
\frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)
|
|
|
|
|
| 5832984291 |
(\sin(x))^2 + (\cos(x))^2 = 1
|
|
|
|
|
| 5857434758 |
\int a dx = a x
|
|
|
|
|
| 5868688585 |
\frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = \frac{p^2}{2m} \psi(\vec{r},t)
|
|
|
|
|
| 5900595848 |
k = \frac{\omega}{v}
|
|
|
|
|
| 5928285821 |
x^2 + 2x(b/(2a)) + (b/(2a))^2 = (x+(b/(2a)))^2
|
|
|
|
|
| 5928292841 |
x^2 + (b/a)x + (b/(2a))^2 = -c/a + (b/(2a))^2
|
|
|
|
|
| 5938459282 |
x^2 + (b/a)x = -c/a
|
|
|
|
|
| 5958392859 |
x^2 + (b/a)x+(c/a) = 0
|
|
|
|
|
| 5959282914 |
x^2 + x(b/a) + (b/(2a))^2 = (x+(b/(2a)))^2
|
|
|
|
|
| 5982958248 |
x = -\sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
|
|
|
|
|
| 5982958249 |
x+(b/(2a)) = -\sqrt{(b/(2a))^2 - (c/a)}
|
|
|
|
|
| 5985371230 |
\vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi(\vec{r},t)
|
|
|
|
|
| 6742123016 |
\vec{p}_{electron}\cdot\vec{p}_{electron} = (\vec{p}_{1}\cdot\vec{p}_{1})+(\vec{p}_{2}\cdot\vec{p}_{2})-2(\vec{p}_{1}\cdot\vec{p}_{2})
|
|
|
|
|
| 7466829492 |
\vec{\nabla} \cdot \vec{E} = 0
|
|
|
|
|
| 7564894985 |
\int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)
|
|
|
|
|
| 7572664728 |
\cos(2x)+2(\sin(x))^2 = 1
|
|
|
|
|
| 7575738420 |
\left(\sin\left(\frac{n \pi}{W}x\right)\right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2}
|
|
|
|
|
| 7575859295 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859300 |
\epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{\nabla} \times \vec{E} )_k = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859302 |
\epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859304 |
\epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h}
|
|
|
|
|
| 7575859306 |
\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859308 |
\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right) = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859310 |
\hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859312 |
\vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7917051060 |
\vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2}
|
|
|
|
|
| 8139187332 |
\vec{p}_{1} = \vec{p}_{2}+\vec{p}_{electron}
|
|
|
|
|
| 8257621077 |
\vec{p}_{before} = \vec{p}_{1}
|
|
|
|
|
| 8311458118 |
\vec{p}_{after} = \vec{p}_{2}+\vec{p}_{electron}
|
|
|
|
|
| 8399484849 |
\langle x^2 -2x\langle x \rangle+\langle x \rangle^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 8484544728 |
-a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x)
|
|
|
|
|
| 8485757728 |
a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(kx)
|
|
|
|
|
| 8485867742 |
\frac{2}{W} = a^2
|
|
|
|
|
| 8489593958 |
d(u v) = u dv + v du
|
|
|
|
|
| 8489593960 |
d(u v) - v du = u dv
|
|
|
|
|
| 8489593962 |
u dv = d(u v) - v du
|
|
|
|
|
| 8489593964 |
\int u dv = u v - \int v du
|
|
|
|
|
| 8494839423 |
\nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
|
|
|
|
|
| 8572657110 |
1 = \int |\psi(x)|^2 dx
|
|
|
|
|
| 8572852424 |
\vec{E} = E(\vec{r},t)
|
|
|
|
|
| 8575746378 |
\int \frac{1}{2} dx = \frac{1}{2} x
|
|
|
|
|
| 8575748999 |
\frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x)\right) = -k^2 \left(a \sin(kx) + b \cos(kx)\right)
|
|
|
|
|
| 8576785890 |
1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
|
|
|
|
|
| 8577275751 |
0 = a \sin(0) + b\cos(0)
|
|
|
|
|
| 8582885111 |
\psi(x) = a \sin(kx) + b \cos(kx)
|
|
|
|
|
| 8582954722 |
x^2 + 2xh + h^2 = (x+h)^2
|
|
|
|
|
| 8849289982 |
\psi(x)^* = a \sin(\frac{n \pi}{W} x)
|
|
|
|
|
| 8889444440 |
1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right)\right)^2 dx
|
|
|
|
|
| 9059289981 |
\psi(x) = a \sin(k x)
|
|
|
|
|
| 9285928292 |
ax^2 + bx + c = 0
|
|
|
|
|
| 9291999979 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0\vec{\nabla} \times \frac{\partial \vec{H}}{\partial t}
|
|
|
|
|
| 9294858532 |
\hat{A}^+ = \hat{A}
|
|
|
|
|
| 9385938295 |
(x+(b/(2a)))^2 = -(c/a) + (b/(2a))^2
|
|
|
|
|
| 9393939991 |
\psi(x) = -\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
|
|
|
|
|
| 9393939992 |
\psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
|
|
|
|
|
| 9394939493 |
\nabla^2 E(\vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r},t)
|
|
|
|
|
| 9429829482 |
\frac{d}{dx} y = -\sin(x) + i\cos(x)
|
|
|
|
|
| 9482113948 |
\frac{dy}{y} = i dx
|
|
|
|
|
| 9482438243 |
(\cos(x))^2 = \cos(2x)+(\sin(x))^2
|
|
|
|
|
| 9482923849 |
\exp(i x) = y
|
|
|
|
|
| 9482928242 |
\cos(2x) = (\cos(x))^2 - (\sin(x))^2
|
|
|
|
|
| 9482928243 |
\cos(2x)+(\sin(x))^2 = (\cos(x))^2
|
|
|
|
|
| 9482943948 |
\log(y) = i dx
|
|
|
|
|
| 9482948292 |
b = c
|
|
|
|
|
| 9482984922 |
\frac{d}{dx} y = (i\sin(x) + \cos(x)) i
|
|
|
|
|
| 9483928192 |
\cos(2x)+i\sin(2x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
|
|
|
|
|
| 9485384858 |
\nabla^2 E(\vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 9485747245 |
\sqrt{\frac{2}{W}} = a
|
|
|
|
|
| 9485747246 |
-\sqrt{\frac{2}{W}} = a
|
|
|
|
|
| 9492920340 |
y = \cos(x)+i \sin(x)
|
|
|
|
|
| 9494829190 |
k
|
|
|
|
|
| 9495857278 |
\psi(x=W) = 0
|
|
|
|
|
| 9499428242 |
E(\vec{r},t) = E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 9499959299 |
a + k = b + k
|
|
|
|
|
| 9582958293 |
x = \sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
|
|
|
|
|
| 9582958294 |
x+(b/(2a)) = \sqrt{(b/(2a))^2 - (c/a)}
|
|
|
|
|
| 9584821911 |
c + d = a
|
|
|
|
|
| 9585727710 |
\psi(x=0) = 0
|
|
|
|
|
| 9596004948 |
x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle
|
|
|
|
|
| 9848292229 |
dy = y i dx
|
|
|
|
|
| 9848294829 |
\frac{d}{dx} y = y i
|
|
|
|
|
| 9858028950 |
\frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
|
|
|
|
|
| 9889984281 |
2(\sin(x))^2 = 1-\cos(2x)
|
|
|
|
|
| 9919999981 |
\rho = 0
|
|
|
|
|
| 9958485859 |
\frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
|
|
|
|
|
| 9988949211 |
(\sin(x))^2 = \frac{1-\cos(2x)}{2}
|
|
|
|
|
| 9991999979 |
\vec{\nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t}
|
|
|
|
|
| 9999998870 |
\frac{\vec{p}}{\hbar} = \vec{k}
|
|
|
|
|
| 9999999870 |
\frac{p}{\hbar} = k
|
|
|
|
|
| 9999999950 |
\beta = 1/(k_{Boltzmann}T)
|
|
|
|
|
| 9999999951 |
\langle x | k \rangle = \frac{\exp(i k x)}{\sqrt{2\pi}}
|
|
|
|
|
| 9999999952 |
f(x) \delta(x-a) = f(a)
|
|
|
|
|
| 9999999953 |
\int_{-\infty}^{\infty} \delta(x) dx = 1
|
|
|
|
|
| 9999999954 |
c = 1/\sqrt{\epsilon_0 \mu_0}
|
|
|
|
|
| 9999999955 |
\vec{E} = -\vec{\nabla} \Psi
|
|
|
|
|
| 9999999956 |
\vec{F} = \frac{\partial}{\partial t}\vec{p}
|
|
|
|
|
| 9999999957 |
\vec{F} = -\vec{\nabla} V
|
|
|
|
|
| 9999999960 |
\hbar = h/(2 \pi)
|
|
|
|
|
| 9999999961 |
\frac{E}{\hbar} = \omega
|
|
|
|
|
| 9999999962 |
p = \hbar k
|
|
|
|
|
| 9999999963 |
\lambda = h/p
|
|
|
|
|
| 9999999964 |
\omega = c k
|
|
|
|
|
| 9999999965 |
E = \omega \hbar
|
|
|
|
|
| 9999999966 |
\vec{L} = \vec{r}\times\vec{p}
|
|
|
|
|
| 9999999967 |
\vec{S} = \frac{1}{\mu_0} \vec{E}\times \vec{B}
|
|
|
|
|
| 9999999968 |
x = \frac{-b-\sqrt{b^2-4ac}}{2a}
|
|
|
|
|
| 9999999969 |
x = \frac{-b+\sqrt{b^2-4ac}}{2a}
|
|
|
|
|
| 9999999970 |
\eta_1 \sin\theta_1 = \eta_2 \sin\theta_2
|
|
|
|
|
| 9999999971 |
{\cal H} = \frac{p^2}{2m}+V
|
|
|
|
|
| 9999999972 |
{\cal H} |\psi\rangle = E |\psi\rangle
|
|
|
|
|
| 9999999973 |
\left( \Delta A \right)^2 = \langle A^2 \rangle - \langle A \rangle^2
|
|
|
|
|
| 9999999974 |
\langle \psi| \hat{A} |\psi\rangle = \int_{-\infty}^{\infty} \psi^* A \psi dx
|
|
|
|
|
| 9999999975 |
\langle \psi| \hat{A} |\psi\rangle = \langle a \rangle
|
|
|
|
|
| 9999999976 |
\hat{p} = -i \hbar \frac{\partial }{\partial x}
|
|
|
|
|
| 9999999977 |
[\hat{x},\hat{p}] = i \hbar
|
|
|
|
|
| 9999999978 |
\vec{\nabla} \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial E}{\partial t}
|
|
|
|
|
| 9999999979 |
\vec{\nabla} \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}
|
|
|
|
|
| 9999999980 |
\vec{\nabla} \cdot \vec{B} = 0
|
|
|
|
|
| 9999999981 |
\vec{\nabla} \cdot \vec{E} = \rho/\epsilon_0
|
|
|
|
|
| 9999999982 |
V = I R + Q/C + L \frac{\partial I}{\partial t}
|
|
|
|
|
| 9999999983 |
C = V A/d
|
|
|
|
|
| 9999999984 |
Q = C V
|
|
|
|
|
| 9999999985 |
V = I R
|
|
|
|
|
| 9999999986 |
\left[\right]\left[\right] = \left[\right]
|
|
|
|
|
| 9999999987 |
1 atmosphere = 101.325 Pascal
|
|
|
|
|
| 9999999988 |
1 atmosphere = 14.7 pounds/(inch^2)
|
|
|
|
|
| 9999999989 |
mass_{electron} = 511000 electronVolts/(q^2)
|
|
|
|
|
| 9999999990 |
Tesla = 10000*Gauss
|
|
|
|
|
| 9999999991 |
Pascal = Newton / (meter^2)
|
|
|
|
|
| 9999999992 |
Tesla = Newton / (Ampere*meter)
|
|
|
|
|
| 9999999993 |
Farad = Coulumb / Volt
|
|
|
|
|
| 9999999995 |
Volt = Joule / Coulumb
|
|
|
|
|
| 9999999996 |
Coulomb = Ampere / second
|
|
|
|
|
| 9999999997 |
Watt = Joule / second
|
|
|
|
|
| 9999999998 |
Joule = Newton*meter
|
|
|
|
|
| 9999999999 |
Newton = kilogram*meter/(second^2)
|
|
|
|
|
| 4961662865 |
x
|
|
|
|
|
| 9110536742 |
2 x
|
|
|
|
|
| 8483686863 |
\sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x)\right)
|
|
|
|
|
| 3470587782 |
\sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
|
|
|
|
|
| 8642992037 |
2
|
|
|
|
|
| 9894826550 |
2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \left(\exp(i x)+\exp(-i x)\right)
|
|
|
|
|
| 4257214632 |
\frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)
|
|
|
|
|
| 8699789241 |
2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)
|
|
|
|
|
| 9180861128 |
2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)
|
|
|
|
|
| 2405307372 |
\sin(2 x) = 2 \sin(x) \cos(x)
|
|
|
|
|
| 3268645065 |
x
|
|
|
|
|
| 9350663581 |
\pi
|
|
|
|
|
| 8332931442 |
\exp(i \pi) = \cos(\pi)+i \sin(\pi)
|
|
|
|
|
| 6885625907 |
\exp(i \pi) = -1 + i 0
|
|
|
|
|
| 3331824625 |
\exp(i \pi) = -1
|
|
|
|
|
| 4901237716 |
1
|
|
|
|
|
| 2501591100 |
\exp(i \pi) + 1 = 0
|
|
|
|
|
| 5136652623 |
E = KE + PE
|
|
mechanical energy is the sum of the potential plus kinetic energies |
|
|
| 7915321076 |
E, KE, PE
|
|
|
|
|
| 3192374582 |
E_2, KE_2, PE_2
|
|
|
|
|
| 7875206161 |
E_2 = KE_2 + PE_2
|
|
|
|
|
| 4229092186 |
E, KE, PE
|
|
|
|
|
| 1490821948 |
E_1, KE_1, PE_1
|
|
|
|
|
| 4303372136 |
E_1 = KE_1 + PE_1
|
|
|
|
|
| 5514556106 |
E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1)
|
|
|
|
|
| 8357234146 |
KE = \frac{1}{2} m v^2
|
|
kinetic energy |
Kinetic_energy
|
|
| 3658066792 |
KE_2, v_2
|
|
|
|
|
| 8082557839 |
KE, v
|
|
|
|
|
| 7676652285 |
KE_2 = \frac{1}{2} m v_2^2
|
|
|
|
|
| 2790304597 |
KE_1, v_1
|
|
|
|
|
| 9880327189 |
KE, v
|
|
|
|
|
| 4928007622 |
KE_1 = \frac{1}{2} m v_1^2
|
|
|
|
|
| 5733146966 |
KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right)
|
|
|
|
|
| 6554292307 |
t
|
|
|
|
|
| 4270680309 |
\frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t}
|
|
|
|
|
| 5781981178 |
x^2 - y^2 = (x+y)(x-y)
|
|
difference of squares |
Difference_of_two_squares
|
|
| 5946759357 |
v_2, v_1
|
|
|
|
|
| 6675701064 |
x, y
|
|
|
|
|
| 4648451961 |
v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1)
|
|
|
|
|
| 9356924046 |
\frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t}
|
|
|
|
|
| 9397152918 |
v = \frac{v_1 + v_2}{2}
|
|
average velocity |
|
|
| 2857430695 |
a = \frac{v_2 - v_1}{t}
|
|
acceleration |
|
|
| 7735737409 |
\frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t}
|
|
|
|
|
| 4784793837 |
\frac{KE_2 - KE_1}{t} = m v a
|
|
|
|
|
| 5345738321 |
F = m a
|
|
Newton's second law of motion |
Newton%27s_laws_of_motion#Newton's_second_law
|
|
| 2186083170 |
\frac{KE_2 - KE_1}{t} = v F
|
|
|
|
|
| 6715248283 |
PE = -F x
|
|
potential energy |
Potential_energy
|
|
| 3852078149 |
PE_2, x_2
|
|
|
|
|
| 7388921188 |
PE, x
|
|
|
|
|
| 2431507955 |
PE_2 = -F x_2
|
|
|
|
|
| 3412641898 |
PE_1, x_1
|
|
|
|
|
| 9937169749 |
PE, x
|
|
|
|
|
| 4669290568 |
PE_1 = -F x_1
|
|
|
|
|
| 7734996511 |
PE_2 - PE_1 = -F ( x_2 - x_1 )
|
|
|
|
|
| 2016063530 |
t
|
|
|
|
|
| 7267155233 |
\frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right)
|
|
|
|
|
| 9337785146 |
v = \frac{x_2 - x_1}{t}
|
|
average velocity |
|
|
| 4872970974 |
\frac{PE_2 - PE_1}{t} = -F v
|
|
|
|
|
| 2081689540 |
t
|
|
|
|
|
| 2770069250 |
\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t}
|
|
|
|
|
| 3591237106 |
\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v
|
|
|
|
|
| 1772416655 |
\frac{E_2 - E_1}{t} = v F - F v
|
|
|
|
|
| 1809909100 |
\frac{E_2 - E_1}{t} = 0
|
|
|
|
|
| 5778176146 |
t
|
|
|
|
|
| 3806977900 |
E_2 - E_1 = 0
|
|
|
|
|
| 5960438249 |
E_1
|
|
|
|
|
| 8558338742 |
E_2 = E_1
|
|
|
|
|
| 8945218208 |
\theta_{Brewster} + \theta_{refracted} = 90^{\circ}
|
|
|
based on figure 34-27 on page 824 in 2001_HRW
|
|
| 9025853427 |
\theta_{Brewster}
|
|
|
|
|
| 1310571337 |
\theta_{refracted} = 90^{\circ} - \theta_{Brewster}
|
|
|
|
|
| 6450985774 |
n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 )
|
|
Law of Refraction |
eq 34-44 on page 819 in 2001_HRW
|
|
| 1779497048 |
\theta_{Brewster}, \theta_{refraction}
|
|
|
|
|
| 7322344455 |
\theta_1, \theta_2
|
|
|
|
|
| 2575937347 |
n_1 \sin( \theta_{Brewster} ) = n_2 \sin( \theta_{refraction} )
|
|
|
|
|
| 7696214507 |
n_1 \sin( \theta_{Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{Brewster} )
|
|
|
|
|
| 8588429722 |
\sin( 90^{\circ} - x ) = \cos( x )
|
|
|
|
|
| 7375348852 |
\theta_{Brewster}
|
|
|
|
|
| 1512581563 |
x
|
|
|
|
|
| 6831637424 |
\sin( 90^{\circ} - \theta_{Brewster} ) = \cos( \theta_{Brewster} )
|
|
|
|
|
| 3061811650 |
n_1 \sin( \theta_{Brewster} ) = n_2 \cos( \theta_{Brewster} )
|
|
|
|
|
| 7857757625 |
n_1
|
|
|
|
|
| 9756089533 |
\sin( \theta_{Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{Brewster} )
|
|
|
|
|
| 5632428182 |
\cos( \theta_{Brewster} )
|
|
|
|
|
| 2768857871 |
\frac{\sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )} = \frac{n_2}{n_1}
|
|
|
|
|
| 4968680693 |
\tan( x ) = \frac{ \sin( x )}{\cos( x )}
|
|
|
|
|
| 7321695558 |
\theta_{Brewster}
|
|
|
|
|
| 9906920183 |
x
|
|
|
|
|
| 4501377629 |
\tan( \theta_{Brewster} ) = \frac{ \sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )}
|
|
|
|
|
| 3417126140 |
\tan( \theta_{Brewster} ) = \frac{ n_2 }{ n_1 }
|
|
|
|
|
| 5453995431 |
\arctan{ x }
|
|
|
|
|
| 6023986360 |
x
|
|
|
|
|
| 8495187962 |
\theta_{Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }
|
|
|
|
|
| 9040079362 |
f
|
|
|
|
|
| 9565166889 |
T
|
|
|
|
|
| 5426308937 |
v = \frac{d}{t}
|
|
|
|
|
| 7476820482 |
C
|
|
|
|
|
| 1277713901 |
d
|
|
|
|
|
| 6946088325 |
v = \frac{C}{t}
|
|
|
|
|
| 6785303857 |
C = 2 \pi r
|
|
|
|
|
| 4816195267 |
C, r
|
|
|
|
|
| 2113447367 |
C_{\rm{Earth\ orbit}}, r_{\rm{Earth\ orbit}}
|
|
|
|
|
| 6348260313 |
C_{\rm{Earth\ orbit}} = 2 \pi r_{\rm{Earth\ orbit}}
|
|
|
|
|
| 3847777611 |
C, v, t
|
|
|
|
|
| 6406487188 |
C_{\rm{Earth\ orbit}}, v_{\rm{Earth\ orbit}}, t_{\rm{Earth\ orbit}}
|
|
|
|
|
| 3046191961 |
v_{\rm{Earth\ orbit}} = \frac{C_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
|
|
|
|
|
| 3080027960 |
v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
|
|
|
|
|
| 7175416299 |
t_{\rm{Earth\ orbit}} = 1 {\rm{year}}
|
|
|
|
|
| 3219318145 |
\frac{365 {\rm days}}{1 {\rm year}} \frac{24 {\rm hours}}{1 {\rm day}} \frac{60 {\rm minutes}}{1 {\rm hour}} \frac{60 {\rm seconds}}{1 {\rm minute}}
|
|
|
|
|
| 8721295221 |
t_{\rm{Earth\ orbit}} = 3.16 10^7 {\rm{seconds}}
|
|
|
|
|
| 4593428198 |
v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{3.16\ 10^7 {\rm seconds}}
|
|
|
|
|
| 3472836147 |
r_{\rm{Earth\ orbit}} = 1.496\ 10^8 {\rm km}
|
|
|
|
|
| 6998364753 |
v_{\rm{Earth\ orbit}} = \frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}}
|
|
|
|
|
| 4180845508 |
v_{\rm{Earth\ orbit}} = 29.8 \frac{{\rm km}}{{\rm sec}}
|
|
|
|
|
| 4662369843 |
x' = \gamma (x - v t)
|
|
|
|
|
| 2983053062 |
x = \gamma (x' + v t')
|
|
|
|
|
| 3426941928 |
x = \gamma ( \gamma (x - v t) + v t' )
|
|
|
|
|
| 2096918413 |
x = \gamma ( \gamma x - \gamma v t + v t' )
|
|
|
|
|
| 7741202861 |
x = \gamma^2 x - \gamma^2 v t + \gamma v t'
|
|
|
|
|
| 7337056406 |
\gamma^2 x
|
|
|
|
|
| 4139999399 |
x - \gamma^2 x = - \gamma^2 v t + \gamma v t'
|
|
|
|
|
| 3495403335 |
x
|
|
|
|
|
| 9031609275 |
x (1 - \gamma^2 ) = - \gamma^2 v t + \gamma v t'
|
|
|
|
|
| 8014566709 |
\gamma^2 v t
|
|
|
|
|
| 9409776983 |
x (1 - \gamma^2 ) + \gamma^2 v t = \gamma v t'
|
|
|
|
|
| 2226340358 |
\gamma v
|
|
|
|
|
| 6417705759 |
\frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = \gamma v t'
|
|
|
|
|
| 8730201316 |
\frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t = t'
|
|
|
first term was multiplied by \gamma/\gamma
|
|
| 1974334644 |
\frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = t'
|
|
|
|
|
| 5148266645 |
t' = \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t
|
|
|
|
|
| 4287102261 |
x^2 + y^2 + z^2 = c^2 t^2
|
|
|
describes a spherical wavefront
|
|
| 1201689765 |
x'^2 + y'^2 + z'^2 = c^2 t'^2
|
|
|
describes a spherical wavefront for an observer in a moving frame of reference
|
|
| 7057864873 |
y' = y
|
|
|
frame of reference is moving only along x direction
|
|
| 8515803375 |
z' = z
|
|
|
frame of reference is moving only along x direction
|
|
| 6153463771 |
3
|
|
|
|
|
| 4746576736 |
asdf
|
|
|
|
|
| 1027270623 |
minga
|
|
|
|
|
| 6101803533 |
ming
|
|
|
|
|
| 9231495607 |
a = b
|
|
|
|
|
| 8664264194 |
b + 2
|
|
|
|
|
| 9805063945 |
\gamma^2 (x - v t)^2 + y^2 + z^2 = c^2 \gamma^2 \left( t + \frac{ 1 - \gamma^2 }{ \gamma^2 } \frac{x}{v} \right)^2
|
|
|
|
|
| 4057686137 |
C \to C_{\rm{Earth\ orbit}}
|
|
|
|
|
| 2346150725 |
r \to r_{\rm{Earth\ orbit}}
|
|
|
|
|
| 9753878784 |
v \to v_{\rm{Earth\ orbit}}
|
|
|
|
|
| 8135396036 |
t \to t_{\rm{Earth\ orbit}}
|
|
|
|
|
| 2773628333 |
\theta_1 \to \theta_{Brewster}
|
|
|
|
|
| 7154592211 |
\theta_2 \to \theta_{\rm refracted}
|
|
|
|
|
| 3749492596 |
E \to E_1
|
|
|
|
|
| 1258245373 |
E \to E_2
|
|
|
|
|
| 4147101187 |
KE \to KE_1
|
|
|
|
|
| 3809726424 |
PE \to PE_1
|
|
|
|
|
| 6383056612 |
KE \to KE_2
|
|
|
|
|
| 5075406409 |
PE \to PE_2
|
|
|
|
|
| 5398681503 |
v \to v_1
|
|
|
|
|
| 9305761407 |
v \to v_2
|
|
|
|
|
| 4218009993 |
x \to x_1
|
|
|
|
|
| 4188639044 |
x \to x_2
|
|
|
|
|
| 8173074178 |
x \to v_2
|
|
|
|
|
| 1025759423 |
y \to v_1
|
|
|
|
|
| 1935543849 |
\gamma^2 x^2 - \gamma^2 2 x v t + \gamma^2 v^2 t^2 + y^2 + z^2 = c^2 \gamma^2 \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x^2}{\gamma^2} + c^2 \gamma^2 2 t \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x}{\gamma} + c^2 \gamma^2 t^2
|
|
|
|
|
| 1586866563 |
\left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right) = t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)
|
|
|
|
|
| 3182633789 |
\gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1
|
|
|
|
|
| 1916173354 |
-\gamma^2 v^2 + c^2 \gamma^2 = c^2
|
|
|
|
|
| 2076171250 |
-\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} = 0
|
|
|
|
|
| 5284610349 |
\gamma^2
|
|
|
|
|
| 2417941373 |
- c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 - \gamma^2
|
|
|
|
|
| 5787469164 |
1 - \gamma^2
|
|
|
|
|
| 1639827492 |
- c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2} = 1
|
|
|
|
|
| 5669500954 |
v^2 \gamma^2
|
|
|
|
|
| 5763749235 |
-c^2 + c^2 \gamma^2 = v^2 \gamma^2
|
|
|
|
|
| 6408214498 |
c^2
|
|
|
|
|
| 2999795755 |
c^2 \gamma^2 = v^2 \gamma^2 + c^2
|
|
|
|
|
| 3412946408 |
v^2 \gamma^2
|
|
|
|
|
| 2542420160 |
c^2 \gamma^2 - v^2 \gamma^2 = c^2
|
|
|
|
|
| 7743841045 |
\gamma^2
|
|
|
|
|
| 7513513483 |
\gamma^2 (c^2 - v^2) = c^2
|
|
|
|
|
| 8571466509 |
c^2 - \gamma^2
|
|
|
|
|
| 7906112355 |
\gamma^2 = \frac{c^2}{c^2 - \gamma^2}
|
|
|
|
|
| 3060139639 |
1/2
|
|
|
|
|
| 1528310784 |
\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
|
|
|
|
|
| 8360117126 |
\gamma = \frac{-1}{\sqrt{1-\frac{v^2}{c^2}}}
|
|
|
not a physically valid result in this context
|
|
| 3366703541 |
a = \frac{v - v_0}{t}
|
|
|
acceleration is the average change in speed over a duration
|
|
| 7083390553 |
t
|
|
|
|
|
| 4748157455 |
a t = v - v_0
|
|
|
|
|
| 6417359412 |
v_0
|
|
|
|
|
| 4798787814 |
a t + v_0 = v
|
|
|
|
|
| 3462972452 |
v = v_0 + a t
|
|
|
|
|
| 3411994811 |
v_{\rm ave} = \frac{d}{t}
|
|
|
|
|
| 6175547907 |
v_{\rm ave} = \frac{v + v_0}{2}
|
|
|
|
|
| 9897284307 |
\frac{d}{t} = \frac{v + v_0}{2}
|
|
|
|
|
| 8865085668 |
t
|
|
|
|
|
| 8706092970 |
d = \left(\frac{v + v_0}{2}\right)t
|
|
|
|
|
| 7011114072 |
d = \frac{(v_0 + a t) + v_0}{2} t
|
|
|
|
|
| 1265150401 |
d = \frac{2 v_0 + a t}{2} t
|
|
|
|
|
| 9658195023 |
d = v_0 t + \frac{1}{2} a t^2
|
|
|
|
|
| 5799753649 |
2
|
|
|
|
|
| 7215099603 |
v^2 = v_0^2 + 2 a t v_0 + a^2 t^2
|
|
|
|
|
| 5144263777 |
v^2 = v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right)
|
|
|
|
|
| 7939765107 |
v^2 = v_0^2 + 2 a d
|
|
|
|
|
| 6729698807 |
v_0
|
|
|
|
|
| 9759901995 |
v - v_0 = a t
|
|
|
|
|
| 2242144313 |
a
|
|
|
|
|
| 1967582749 |
t = \frac{v - v_0}{a}
|
|
|
|
|
| 5733721198 |
d = \frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right)
|
|
|
|
|
| 5611024898 |
d = \frac{1}{2 a} (v^2 - v_0^2)
|
|
|
|
|
| 5542390646 |
2 a
|
|
|
|
|
| 8269198922 |
2 a d = v^2 - v_0^2
|
|
|
|
|
| 9070454719 |
v_0^2
|
|
|
|
|
| 4948763856 |
2 a d + v_0^2 = v^2
|
|
|
|
|
| 9645178657 |
a t
|
|
|
|
|
| 6457044853 |
v - a t = v_0
|
|
|
|
|
| 1259826355 |
d = (v - a t) t + \frac{1}{2} a t^2
|
|
|
|
|
| 4580545876 |
d = v t - a t^2 + \frac{1}{2} a t^2
|
|
|
|
|
| 6421241247 |
d = v t - \frac{1}{2} a t^2
|
|
|
|
|
| 4182362050 |
Z = |Z| \exp( i \theta )
|
|
|
Z \in \mathbb{C}
|
|
| 1928085940 |
Z^* = |Z| \exp( -i \theta )
|
|
|
|
|
| 9191880568 |
Z Z^* = |Z| |Z| \exp( -i \theta ) \exp( i \theta )
|
|
|
|
|
| 3350830826 |
Z Z^* = |Z|^2
|
|
|
|
|
| 8396997949 |
I = | A + B |^2
|
|
|
intensity of two waves traveling opposite directions on same path
|
|
| 4437214608 |
Z
|
|
|
|
|
| 5623794884 |
A + B
|
|
|
|
|
| 2236639474 |
(A + B)(A + B)^* = |A + B|^2
|
|
|
|
|
| 1020854560 |
I = (A + B)(A + B)^*
|
|
|
|
|
| 6306552185 |
I = (A + B)(A^* + B^*)
|
|
|
|
|
| 8065128065 |
I = A A^* + B B^* + A B^* + B A^*
|
|
|
|
|
| 9761485403 |
Z
|
|
|
|
|
| 8710504862 |
A
|
|
|
|
|
| 4075539836 |
A A^* = |A|^2
|
|
|
|
|
| 6529120965 |
B
|
|
|
|
|
| 1511199318 |
Z
|
|
|
|
|
| 7107090465 |
B B^* = |B|^2
|
|
|
|
|
| 5125940051 |
I = |A|^2 + B B^* + A B^* + B A^*
|
|
|
|
|
| 1525861537 |
I = |A|^2 + |B|^2 + A B^* + B A^*
|
|
|
|
|
| 1742775076 |
Z \to B
|
|
|
|
|
| 7607271250 |
\theta \to \phi
|
|
|
|
|
| 4192519596 |
B = |B| \exp(i \phi)
|
|
|
|
|
| 2064205392 |
A
|
|
|
|
|
| 1894894315 |
Z
|
|
|
|
|
| 1357848476 |
A = |A| \exp(i \theta)
|
|
|
|
|
| 6555185548 |
A^* = |A| \exp(-i \theta)
|
|
|
|
|
| 4504256452 |
B^* = |B| \exp(-i \phi)
|
|
|
|
|
| 7621705408 |
I = |A|^2 + |B|^2 + |A| |B| \exp(-i \theta) \exp(i \phi) + |A| |B| \exp(i \theta) \exp(-i \phi)
|
|
|
|
|
| 3085575328 |
I = |A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi))
|
|
|
|
|
| 4935235303 |
x
|
|
|
|
|
| 2293352649 |
\theta - \phi
|
|
|
|
|
| 3660957533 |
\cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
|
|
|
|
|
| 3967985562 |
2
|
|
|
|
|
| 2700934933 |
2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
|
|
|
|
|
| 8497631728 |
I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )
|
|
|
|
|
| 6774684564 |
\theta = \phi
|
|
|
for coherent waves
|
|
| 2099546947 |
I = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )
|
|
|
|
|
| 2719691582 |
|A| = |B|
|
|
|
in a loop
|
|
| 3257276368 |
I = |A|^2 + |A|^2 + |A| |A| 2
|
|
|
|
|
| 8283354808 |
I_{\rm coherent} = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )
|
|
|
|
|
| 8046208134 |
I_{\rm coherent} = |A|^2 + |A|^2 + |A| |A| 2
|
|
|
|
|
| 1172039918 |
I_{\rm coherent} = 4 |A|^2
|
|
|
|
|
| 8602221482 |
\langle \cos(\theta - \phi) \rangle = 0
|
|
|
incoherent light source
|
|
| 6240206408 |
I_{\rm incoherent} = |A|^2 + |B|^2
|
|
|
|
|
| 6529793063 |
I_{\rm incoherent} = |A|^2 + |A|^2
|
|
|
|
|
| 3060393541 |
I_{\rm incoherent} = 2|A|^2
|
|
|
|
|
| 6556875579 |
\frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2
|
|
|
|
|
| 8750500372 |
f
|
|
|
|
|
| 7543102525 |
g
|
|
|
|
|
| 8267133829 |
a = b
|
|
|
|
|
| 7968822469 |
c = d
|
|
|
|
|
| 3169580383 |
\vec{a} = \frac{d\vec{v}}{dt}
|
|
|
acceleration is the change in speed over a duration
|
|
| 8602512487 |
\vec{a} = a_x \hat{x} + a_y \hat{y}
|
|
|
decompose acceleration into two components
|
|
| 4158986868 |
a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt}
|
|
|
|
|
| 5349866551 |
\vec{v} = v_x \hat{x} + v_y \hat{y}
|
|
|
|
|
| 7729413831 |
a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right)
|
|
|
|
|
| 1819663717 |
a_x = \frac{d}{dt} v_x
|
|
|
|
|
| 8228733125 |
a_y = \frac{d}{dt} v_y
|
|
|
|
|
| 9707028061 |
a_x = 0
|
|
|
|
|
| 2741489181 |
a_y = -g
|
|
|
|
|
| 3270039798 |
2
|
|
|
|
|
| 8880467139 |
2
|
|
|
|
|
| 8750379055 |
0 = \frac{d}{dt} v_x
|
|
|
|
|
| 1977955751 |
-g = \frac{d}{dt} v_y
|
|
|
|
|
| 6672141531 |
dt
|
|
|
|
|
| 1702349646 |
-g dt = d v_y
|
|
|
|
|
| 8584698994 |
-g \int dt = \int d v_y
|
|
|
|
|
| 9973952056 |
-g t = v_y - v_{0, y}
|
|
|
|
|
| 4167526462 |
v_{0, y}
|
|
|
|
|
| 6572039835 |
-g t + v_{0, y} = v_y
|
|
|
|
|
| 7252338326 |
v_y = \frac{dy}{dt}
|
|
|
|
|
| 6204539227 |
-g t + v_{0, y} = \frac{dy}{dt}
|
|
|
|
|
| 1614343171 |
dt
|
|
|
|
|
| 8145337879 |
-g t dt + v_{0, y} dt = dy
|
|
|
|
|
| 8808860551 |
-g \int t dt + v_{0, y} \int dt = \int dy
|
|
|
|
|
| 2858549874 |
- \frac{1}{2} g t^2 + v_{0, y} t = y - y_0
|
|
|
|
|
| 6098638221 |
y_0
|
|
|
|
|
| 2461349007 |
- \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y
|
|
|
|
|
| 8717193282 |
dt
|
|
|
|
|
| 1166310428 |
0 dt = d v_x
|
|
|
|
|
| 2366691988 |
\int 0 dt = \int d v_x
|
|
|
|
|
| 1676472948 |
0 = v_x - v_{0, x}
|
|
|
|
|
| 1439089569 |
v_{0, x}
|
|
|
|
|
| 6134836751 |
v_{0, x} = v_x
|
|
|
|
|
| 8460820419 |
v_x = \frac{dx}{dt}
|
|
|
|
|
| 7455581657 |
v_{0, x} = \frac{dx}{dt}
|
|
|
|
|
| 8607458157 |
dt
|
|
|
|
|
| 1963253044 |
v_{0, x} dt = dx
|
|
|
|
|
| 3676159007 |
v_{0, x} \int dt = \int dx
|
|
|
|
|
| 9882526611 |
v_{0, x} t = x - x_0
|
|
|
|
|
| 3182907803 |
x_0
|
|
|
|
|
| 8486706976 |
v_{0, x} t + x_0 = x
|
|
|
|
|
| 1306360899 |
x = v_{0, x} t + x_0
|
|
|
|
|
| 7049769409 |
2
|
|
|
|
|
| 9341391925 |
\vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y}
|
|
|
|
|
| 6410818363 |
\theta
|
|
|
|
|
| 7391837535 |
\cos(\theta) = \frac{v_{0, x}}{v_0}
|
|
|
|
|
| 7376526845 |
\sin(\theta) = \frac{v_{0, y}}{v_0}
|
|
|
|
|
| 5868731041 |
v_0
|
|
|
|
|
| 6083821265 |
v_0 \cos(\theta) = v_{0, x}
|
|
|
|
|
| 5438722682 |
x = v_0 t \cos(\theta) + x_0
|
|
|
|
|
| 5620558729 |
v_0
|
|
|
|
|
| 8949329361 |
v_0 \sin(\theta) = v_{0, y}
|
|
|
|
|
| 1405465835 |
y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0
|
|
|
|
|
| 9862900242 |
y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0
|
|
|
|
|
| 6050070428 |
v_{0, x}
|
|
|
|
|
| 3274926090 |
t = \frac{x - x_0}{v_{0, x}}
|
|
|
|
|
| 7354529102 |
y = - \frac{1}{2} g \left( \frac{x - x_0}{v_{0, x}} \right)^2 + v_{0, y} \frac{x - x_0}{v_{0, x}} + y_0
|
|
|
|
|
| 8406170337 |
y \to y_f
|
|
|
|
|
| 2403773761 |
t \to t_f
|
|
|
|
|
| 5379546684 |
y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
|
|
|
|
|
| 5373931751 |
t = t_f
|
|
|
|
|
| 9112191201 |
y_f = 0
|
|
|
|
|
| 8198310977 |
0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
|
|
|
|
|
| 1650441634 |
y_0 = 0
|
|
|
define coordinate system such that initial height is at origin
|
|
| 1087417579 |
0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)
|
|
|
|
|
| 4829590294 |
t_f
|
|
|
|
|
| 2086924031 |
0 = - \frac{1}{2} g t_f + v_0 \sin(\theta)
|
|
|
|
|
| 6974054946 |
\frac{1}{2} g t_f
|
|
|
|
|
| 1191796961 |
\frac{1}{2} g t_f = v_0 \sin(\theta)
|
|
|
|
|
| 2510804451 |
2/g
|
|
|
|
|
| 4778077984 |
t_f = \frac{2 v_0 \sin(\theta)}{g}
|
|
|
|
|
| 3273630811 |
x \to x_f
|
|
|
|
|
| 6732786762 |
t \to t_f
|
|
|
|
|
| 3485125659 |
x_f = v_0 t_f \cos(\theta) + x_0
|
|
|
|
|
| 4370074654 |
t = t_f
|
|
|
|
|
| 2378095808 |
x_f = x_0 + d
|
|
|
|
|
| 4268085801 |
x_0 + d = v_0 t_f \cos(\theta) + x_0
|
|
|
|
|
| 8072682558 |
x_0
|
|
|
|
|
| 7233558441 |
d = v_0 t_f \cos(\theta)
|
|
|
|
|
| 2297105551 |
d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)
|
|
|
|
|
| 7587034465 |
\theta
|
|
|
|
|
| 7214442790 |
x
|
|
|
|
|
| 2519058903 |
\sin(2 \theta) = 2 \sin(\theta) \cos(\theta)
|
|
|
|
|
| 8922441655 |
d = \frac{v_0^2}{g} \sin(2 \theta)
|
|
|
|
|
| 5667870149 |
\theta
|
|
|
|
|
| 1541916015 |
\theta = \frac{\pi}{4}
|
|
|
|
|
| 3607070319 |
d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)
|
|
|
|
|
| 5353282496 |
d = \frac{v_0^2}{g}
|
|
|
|
|
| 0000040490 |
a^2
|
|
|
|
|
| 0000999900 |
b/(2a)
|
|
|
|
|
| 0001030901 |
\cos(x)
|
|
|
|
|
| 0001111111 |
(\sin(x))^2
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| 0001209482 |
2 \pi
|
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| 0001304952 |
\hbar
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| 0001334112 |
W
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| 0001921933 |
2 i
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| 0002239424 |
2
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| 0002338514 |
\vec{p}_{2}
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| 0002342425 |
m/m
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| 0002367209 |
1/2
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| 0002393922 |
x
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| 0002424922 |
a
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| 0002436656 |
i \hbar
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| 0002449291 |
b/(2a)
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| 0002838490 |
b/(2a)
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| 0002919191 |
\sin(-x)
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| 0002929944 |
1/2
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| 0002940021 |
2 \pi
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| 0003232242 |
t
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| 0003413423 |
\cos(-x)
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| 0003747849 |
-1
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| 0003838111 |
2
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| 0003919391 |
x
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| 0003949052 |
-x
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| 0003949921 |
\hbar
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| 0003954314 |
dx
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| 0003981813 |
-\sin(x)
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| 0004089571 |
2x
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| 0004264724 |
y
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| 0004307451 |
(b/(2a))^2
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| 0004582412 |
x
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| 0004829194 |
2
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| 0004831494 |
a
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| 0004849392 |
x
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| 0004858592 |
h
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| 0004934845 |
x
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| 0004948585 |
a
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| 0005395034 |
a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
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| 0005626421 |
t
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| 0005749291 |
f
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| 0005938585 |
\frac{-\hbar^2}{2m}
|
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| 0006466214 |
(\sin(x))^2
|
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| 0006544644 |
t
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| 0006563727 |
x
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| 0006644853 |
c/a
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| 0006656532 |
e
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| 0007471778 |
2(\sin(x))^2
|
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| 0007563791 |
i
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| 0007636749 |
x
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| 0007894942 |
(\sin(x))^2
|
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| 0008837284 |
T
|
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| 0008842811 |
\cos(2x)
|
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| 0009458842 |
\psi(x)
|
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| 0009484724 |
\frac{n \pi}{W}x
|
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| 0009485857 |
a^2\frac{2}{W}
|
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| 0009485858 |
\frac{2n\pi}{W}
|
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| 0009492929 |
v du
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| 0009587738 |
\psi
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| 0009594995 |
1/2
|
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| 0009877781 |
y
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|
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| 0203024440 |
1 = \int_0^W a \sin(\frac{n \pi}{W} x) \psi(x)^* dx
|
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|
|
|
| 0404050504 |
\lambda = \frac{v}{f}
|
|
|
|
|
| 0439492440 |
\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)\right|_0^W
|
|
|
|
|
| 0934990943 |
k = \frac{2 \pi}{v T}
|
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|
|
| 0948572140 |
\int \cos(a x) dx = \frac{1}{a}\sin(a x)
|
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|
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| 1010393913 |
\langle \psi| \hat{A}^+ |\psi \rangle = \langle a \rangle^*
|
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| 1010393944 |
x = \langle\psi_{\alpha}| a_{\beta} |\psi_{\beta} \rangle
|
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|
|
| 1010923823 |
k W = n \pi
|
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|
|
|
| 1020010291 |
0 = a \sin(k W)
|
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|
|
| 1020394900 |
p = h/\lambda
|
|
|
|
|
| 1020394902 |
E = h f
|
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|
|
| 1029039903 |
p = m v
|
|
|
|
|
| 1029039904 |
p^2 = m^2 v^2
|
|
|
|
|
| 1158485859 |
\frac{-\hbar^2}{2m} \nabla^2 = {\cal H}
|
|
|
|
|
| 1202310110 |
\frac{1}{a^2} = \int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
|
|
|
|
|
| 1202312210 |
\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
|
|
|
|
|
| 1203938249 |
a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle = a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
|
|
|
|
|
| 1248277773 |
\cos(2x) = 1-2(\sin(x))^2
|
|
|
|
|
| 1293913110 |
0 = b
|
|
|
|
|
| 1293923844 |
\lambda = v T
|
|
|
|
|
| 1314464131 |
\vec{\nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
|
|
|
|
|
| 1314864131 |
\vec{\nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E}
|
|
|
|
|
| 1395858355 |
x = \langle \psi_{\alpha}| a_{\alpha} |\psi_{\beta}\rangle
|
|
|
|
|
| 1492811142 |
f = x - d
|
|
|
|
|
| 1492842000 |
\nabla \vec{x} = f(y)
|
|
|
|
|
| 1636453295 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = - \nabla^2 \vec{E}
|
|
|
|
|
| 1638282134 |
\vec{p}_{before} = \vec{p}_{after}
|
|
|
|
|
| 1648958381 |
\nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{\nabla}\psi(\vec{r},t)\right)
|
|
|
|
|
| 1857710291 |
0 = a \sin(n \pi)
|
|
|
|
|
| 1858578388 |
\nabla^2 E(\vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 1858772113 |
k = \frac{n \pi}{W}
|
|
|
|
|
| 1934748140 |
\int |\psi(x)|^2 dx = 1
|
|
|
|
|
| 2029293929 |
\nabla^2 E(\vec{r})\exp(i \omega t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 2103023049 |
\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right)
|
|
|
|
|
| 2113211456 |
f = 1/T
|
|
|
|
|
| 2123139121 |
-\exp(-i x) = -\cos(x)+i \sin(x)
|
|
|
|
|
| 2131616531 |
T f = 1
|
|
|
|
|
| 2258485859 |
{\cal H} \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
|
|
|
|
|
| 2394240499 |
x = a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle
|
|
|
|
|
| 2394853829 |
\exp(-i x) = \cos(-x)+i \sin(-x)
|
|
|
|
|
| 2394935831 |
( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
|
|
|
|
|
| 2394935835 |
\left(\langle\psi| \hat{A} |\psi\rangle \right)^+ = \left(\langle a \rangle\right)^+
|
|
|
|
|
| 2395958385 |
\nabla^2 \psi\left(\vec{r},t)\right) = \frac{-p^2}{\hbar} \psi(\vec{r},t)
|
|
|
|
|
| 2404934990 |
\langle x^2\rangle -2\langle x \rangle\langle x \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 2494533900 |
\langle x^2\rangle -\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 2569154141 |
\vec{\nabla} \times \frac{\partial}{\partial t}\vec{H} = \epsilon_0 \frac{\partial^2 }{\partial t^2}\vec{E}
|
|
|
|
|
| 2648958382 |
\nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p}\psi(\vec{r},t)\right)
|
|
|
|
|
| 2848934890 |
\langle a \rangle^* = \langle a \rangle
|
|
|
|
|
| 2944838499 |
\psi(x) = a \sin(\frac{n \pi}{W} x)
|
|
|
|
|
| 3121234211 |
\frac{k}{2\pi} = \lambda
|
|
|
|
|
| 3121234212 |
p = \frac{h k}{2\pi}
|
|
|
|
|
| 3121513111 |
k = \frac{2 \pi}{\lambda}
|
|
|
|
|
| 3131111133 |
T = 1 / f
|
|
|
|
|
| 3131211131 |
\omega = 2 \pi f
|
|
|
|
|
| 3132131132 |
\omega = \frac{2\pi}{T}
|
|
|
|
|
| 3147472131 |
\frac{\omega}{2 \pi} = f
|
|
|
|
|
| 3285732911 |
(\cos(x))^2 = 1-(\sin(x))^2
|
|
|
|
|
| 3485475729 |
\nabla^2 E(\vec{r}) = - \frac{\omega^2}{c^2} E(\vec{r})
|
|
|
|
|
| 3585845894 |
\langle \left(x-\langle x \rangle\right)^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 3829492824 |
\frac{1}{2}\left(\exp(i x)+\exp(-i x)\right) = \cos(x)
|
|
|
|
|
| 3924948349 |
a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
|
|
|
|
|
| 3942849294 |
\exp(i x)-\exp(-i x) = 2 i \sin(x)
|
|
|
|
|
| 3943939590 |
x = a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle
|
|
|
|
|
| 3947269979 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
|
|
|
|
|
| 3948571256 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi(\vec{r},t)
|
|
|
|
|
| 3948572230 |
\vec{\nabla}\psi(\vec{r},t) = \psi_0 \vec{\nabla}\exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
|
|
|
|
|
| 3948574224 |
\psi(\vec{r},t) = \psi_0 \exp\left(i\left(\vec{k}\cdot\vec{r} - \omega t \right) \right)
|
|
|
|
|
| 3948574226 |
\psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right)
|
|
|
|
|
| 3948574228 |
\psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
|
|
|
|
|
| 3948574230 |
\psi(\vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
|
|
|
|
|
| 3948574233 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
|
|
|
|
|
| 3948574235 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
|
|
|
|
|
| 3951205425 |
\vec{p}_{after} = \vec{p}_{1}
|
|
|
|
|
| 4147472132 |
E = \frac{h \omega}{2 \pi}
|
|
|
|
|
| 4298359835 |
E = \frac{1}{2}m v^2
|
|
|
|
|
| 4298359845 |
E = \frac{1}{2m}m^2 v^2
|
|
|
|
|
| 4298359851 |
E = \frac{p^2}{2m}
|
|
|
|
|
| 4341171256 |
i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{p^2}{2 m} \psi(\vec{r},t)
|
|
|
|
|
| 4348571256 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi(\vec{r},t)
|
|
|
|
|
| 4394958389 |
\vec{\nabla}\cdot \left(\vec{\nabla}\psi(\vec{r},t)\right) = \frac{i}{\hbar} \vec{\nabla}\cdot\left( \vec{p} \psi(\vec{r},t)\right)
|
|
|
|
|
| 4585828572 |
\epsilon_0 \mu_0 = \frac{1}{c^2}
|
|
|
|
|
| 4585932229 |
\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
|
|
|
|
|
| 4598294821 |
\exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
|
|
|
|
|
| 4638429483 |
\exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))
|
|
|
|
|
| 4742644828 |
\exp(i x)+\exp(-i x) = 2 \cos(x)
|
|
|
|
|
| 4827492911 |
\cos(2x)+(\sin(x))^2 = 1-(\sin(x))^2
|
|
|
|
|
| 4838429483 |
\exp(2 i x) = \cos(2 x)+i \sin(2 x)
|
|
|
|
|
| 4843995999 |
\frac{1}{2 i}\left(\exp(i x)-\exp(-i x)\right) = \sin(x)
|
|
|
|
|
| 4857472413 |
1 = \int \psi(x)\psi(x)^* dx
|
|
|
|
|
| 4857475848 |
\frac{1}{a^2} = \frac{W}{2}
|
|
|
|
|
| 4858429483 |
\exp(i x)\exp(i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))
|
|
|
|
|
| 4923339482 |
i x = \log(y)
|
|
|
|
|
| 4928239482 |
\log(y) = i x
|
|
|
|
|
| 4928923942 |
a = b
|
|
|
|
|
| 4929218492 |
a+b = c
|
|
|
|
|
| 4929482992 |
b = 2
|
|
|
|
|
| 4938429482 |
\exp(-i x) = \cos(x)+i \sin(-x)
|
|
|
|
|
| 4938429483 |
\exp(i x) = \cos(x)+i \sin(x)
|
|
|
|
|
| 4938429484 |
\exp(-i x) = \cos(x)-i \sin(x)
|
|
|
|
|
| 4943571230 |
\vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
|
|
|
|
|
| 4948934890 |
\langle \psi| \hat{A} |\psi\rangle = \langle a \rangle^*
|
|
|
|
|
| 4949359835 |
\langle x^2\rangle -2\langle x^2 \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 4954839242 |
\cos(2x)+i\sin(2x) = (\cos(x)+i \sin(x))(\cos(x)+i \sin(x))
|
|
|
|
|
| 4978429483 |
\exp(-i x) = \cos(-x)+i \sin(-x)
|
|
|
|
|
| 4984892984 |
a+2 = c
|
|
|
|
|
| 4985825552 |
\nabla^2 E(\vec{r})\exp(i \omega t) = i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 5530148480 |
\vec{p}_{1}-\vec{p}_{2} = \vec{p}_{electron}
|
|
|
|
|
| 5727578862 |
\frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)
|
|
|
|
|
| 5832984291 |
(\sin(x))^2 + (\cos(x))^2 = 1
|
|
|
|
|
| 5857434758 |
\int a dx = a x
|
|
|
|
|
| 5868688585 |
\frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = \frac{p^2}{2m} \psi(\vec{r},t)
|
|
|
|
|
| 5900595848 |
k = \frac{\omega}{v}
|
|
|
|
|
| 5928285821 |
x^2 + 2x(b/(2a)) + (b/(2a))^2 = (x+(b/(2a)))^2
|
|
|
|
|
| 5928292841 |
x^2 + (b/a)x + (b/(2a))^2 = -c/a + (b/(2a))^2
|
|
|
|
|
| 5938459282 |
x^2 + (b/a)x = -c/a
|
|
|
|
|
| 5958392859 |
x^2 + (b/a)x+(c/a) = 0
|
|
|
|
|
| 5959282914 |
x^2 + x(b/a) + (b/(2a))^2 = (x+(b/(2a)))^2
|
|
|
|
|
| 5982958248 |
x = -\sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
|
|
|
|
|
| 5982958249 |
x+(b/(2a)) = -\sqrt{(b/(2a))^2 - (c/a)}
|
|
|
|
|
| 5985371230 |
\vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi(\vec{r},t)
|
|
|
|
|
| 6742123016 |
\vec{p}_{electron}\cdot\vec{p}_{electron} = (\vec{p}_{1}\cdot\vec{p}_{1})+(\vec{p}_{2}\cdot\vec{p}_{2})-2(\vec{p}_{1}\cdot\vec{p}_{2})
|
|
|
|
|
| 7466829492 |
\vec{\nabla} \cdot \vec{E} = 0
|
|
|
|
|
| 7564894985 |
\int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)
|
|
|
|
|
| 7572664728 |
\cos(2x)+2(\sin(x))^2 = 1
|
|
|
|
|
| 7575738420 |
\left(\sin\left(\frac{n \pi}{W}x\right)\right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2}
|
|
|
|
|
| 7575859295 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859300 |
\epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{\nabla} \times \vec{E} )_k = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859302 |
\epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
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|
|
|
| 7575859304 |
\epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h}
|
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|
| 7575859306 |
\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
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|
|
| 7575859308 |
\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right) = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
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|
| 7575859310 |
\hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
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|
|
|
| 7575859312 |
\vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
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|
|
|
| 7917051060 |
\vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2}
|
|
|
|
|
| 8139187332 |
\vec{p}_{1} = \vec{p}_{2}+\vec{p}_{electron}
|
|
|
|
|
| 8257621077 |
\vec{p}_{before} = \vec{p}_{1}
|
|
|
|
|
| 8311458118 |
\vec{p}_{after} = \vec{p}_{2}+\vec{p}_{electron}
|
|
|
|
|
| 8399484849 |
\langle x^2 -2x\langle x \rangle+\langle x \rangle^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 8484544728 |
-a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x)
|
|
|
|
|
| 8485757728 |
a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(kx)
|
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|
|
|
| 8485867742 |
\frac{2}{W} = a^2
|
|
|
|
|
| 8489593958 |
d(u v) = u dv + v du
|
|
|
|
|
| 8489593960 |
d(u v) - v du = u dv
|
|
|
|
|
| 8489593962 |
u dv = d(u v) - v du
|
|
|
|
|
| 8489593964 |
\int u dv = u v - \int v du
|
|
|
|
|
| 8494839423 |
\nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
|
|
|
|
|
| 8572657110 |
1 = \int |\psi(x)|^2 dx
|
|
|
|
|
| 8572852424 |
\vec{E} = E(\vec{r},t)
|
|
|
|
|
| 8575746378 |
\int \frac{1}{2} dx = \frac{1}{2} x
|
|
|
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|
| 8575748999 |
\frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x)\right) = -k^2 \left(a \sin(kx) + b \cos(kx)\right)
|
|
|
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|
| 8576785890 |
1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
|
|
|
|
|
| 8577275751 |
0 = a \sin(0) + b\cos(0)
|
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|
|
|
| 8582885111 |
\psi(x) = a \sin(kx) + b \cos(kx)
|
|
|
|
|
| 8582954722 |
x^2 + 2xh + h^2 = (x+h)^2
|
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|
|
|
| 8849289982 |
\psi(x)^* = a \sin(\frac{n \pi}{W} x)
|
|
|
|
|
| 8889444440 |
1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right)\right)^2 dx
|
|
|
|
|
| 9059289981 |
\psi(x) = a \sin(k x)
|
|
|
|
|
| 9285928292 |
ax^2 + bx + c = 0
|
|
|
|
|
| 9291999979 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0\vec{\nabla} \times \frac{\partial \vec{H}}{\partial t}
|
|
|
|
|
| 9294858532 |
\hat{A}^+ = \hat{A}
|
|
|
|
|
| 9385938295 |
(x+(b/(2a)))^2 = -(c/a) + (b/(2a))^2
|
|
|
|
|
| 9393939991 |
\psi(x) = -\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
|
|
|
|
|
| 9393939992 |
\psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
|
|
|
|
|
| 9394939493 |
\nabla^2 E(\vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r},t)
|
|
|
|
|
| 9429829482 |
\frac{d}{dx} y = -\sin(x) + i\cos(x)
|
|
|
|
|
| 9482113948 |
\frac{dy}{y} = i dx
|
|
|
|
|
| 9482438243 |
(\cos(x))^2 = \cos(2x)+(\sin(x))^2
|
|
|
|
|
| 9482923849 |
\exp(i x) = y
|
|
|
|
|
| 9482928242 |
\cos(2x) = (\cos(x))^2 - (\sin(x))^2
|
|
|
|
|
| 9482928243 |
\cos(2x)+(\sin(x))^2 = (\cos(x))^2
|
|
|
|
|
| 9482943948 |
\log(y) = i dx
|
|
|
|
|
| 9482948292 |
b = c
|
|
|
|
|
| 9482984922 |
\frac{d}{dx} y = (i\sin(x) + \cos(x)) i
|
|
|
|
|
| 9483928192 |
\cos(2x)+i\sin(2x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
|
|
|
|
|
| 9485384858 |
\nabla^2 E(\vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 9485747245 |
\sqrt{\frac{2}{W}} = a
|
|
|
|
|
| 9485747246 |
-\sqrt{\frac{2}{W}} = a
|
|
|
|
|
| 9492920340 |
y = \cos(x)+i \sin(x)
|
|
|
|
|
| 9494829190 |
k
|
|
|
|
|
| 9495857278 |
\psi(x=W) = 0
|
|
|
|
|
| 9499428242 |
E(\vec{r},t) = E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 9499959299 |
a + k = b + k
|
|
|
|
|
| 9582958293 |
x = \sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
|
|
|
|
|
| 9582958294 |
x+(b/(2a)) = \sqrt{(b/(2a))^2 - (c/a)}
|
|
|
|
|
| 9584821911 |
c + d = a
|
|
|
|
|
| 9585727710 |
\psi(x=0) = 0
|
|
|
|
|
| 9596004948 |
x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle
|
|
|
|
|
| 9848292229 |
dy = y i dx
|
|
|
|
|
| 9848294829 |
\frac{d}{dx} y = y i
|
|
|
|
|
| 9858028950 |
\frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
|
|
|
|
|
| 9889984281 |
2(\sin(x))^2 = 1-\cos(2x)
|
|
|
|
|
| 9919999981 |
\rho = 0
|
|
|
|
|
| 9958485859 |
\frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
|
|
|
|
|
| 9988949211 |
(\sin(x))^2 = \frac{1-\cos(2x)}{2}
|
|
|
|
|
| 9991999979 |
\vec{\nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t}
|
|
|
|
|
| 9999998870 |
\frac{\vec{p}}{\hbar} = \vec{k}
|
|
|
|
|
| 9999999870 |
\frac{p}{\hbar} = k
|
|
|
|
|
| 9999999950 |
\beta = 1/(k_{Boltzmann}T)
|
|
|
|
|
| 9999999951 |
\langle x | k \rangle = \frac{\exp(i k x)}{\sqrt{2\pi}}
|
|
|
|
|
| 9999999952 |
f(x) \delta(x-a) = f(a)
|
|
|
|
|
| 9999999953 |
\int_{-\infty}^{\infty} \delta(x) dx = 1
|
|
|
|
|
| 9999999954 |
c = 1/\sqrt{\epsilon_0 \mu_0}
|
|
|
|
|
| 9999999955 |
\vec{E} = -\vec{\nabla} \Psi
|
|
|
|
|
| 9999999956 |
\vec{F} = \frac{\partial}{\partial t}\vec{p}
|
|
|
|
|
| 9999999957 |
\vec{F} = -\vec{\nabla} V
|
|
|
|
|
| 9999999960 |
\hbar = h/(2 \pi)
|
|
|
|
|
| 9999999961 |
\frac{E}{\hbar} = \omega
|
|
|
|
|
| 9999999962 |
p = \hbar k
|
|
|
|
|
| 9999999963 |
\lambda = h/p
|
|
|
|
|
| 9999999964 |
\omega = c k
|
|
|
|
|
| 9999999965 |
E = \omega \hbar
|
|
|
|
|
| 9999999966 |
\vec{L} = \vec{r}\times\vec{p}
|
|
|
|
|
| 9999999967 |
\vec{S} = \frac{1}{\mu_0} \vec{E}\times \vec{B}
|
|
|
|
|
| 9999999968 |
x = \frac{-b-\sqrt{b^2-4ac}}{2a}
|
|
|
|
|
| 9999999969 |
x = \frac{-b+\sqrt{b^2-4ac}}{2a}
|
|
|
|
|
| 9999999970 |
\eta_1 \sin\theta_1 = \eta_2 \sin\theta_2
|
|
|
|
|
| 9999999971 |
{\cal H} = \frac{p^2}{2m}+V
|
|
|
|
|
| 9999999972 |
{\cal H} |\psi\rangle = E |\psi\rangle
|
|
|
|
|
| 9999999973 |
\left( \Delta A \right)^2 = \langle A^2 \rangle - \langle A \rangle^2
|
|
|
|
|
| 9999999974 |
\langle \psi| \hat{A} |\psi\rangle = \int_{-\infty}^{\infty} \psi^* A \psi dx
|
|
|
|
|
| 9999999975 |
\langle \psi| \hat{A} |\psi\rangle = \langle a \rangle
|
|
|
|
|
| 9999999976 |
\hat{p} = -i \hbar \frac{\partial }{\partial x}
|
|
|
|
|
| 9999999977 |
[\hat{x},\hat{p}] = i \hbar
|
|
|
|
|
| 9999999978 |
\vec{\nabla} \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial E}{\partial t}
|
|
|
|
|
| 9999999979 |
\vec{\nabla} \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}
|
|
|
|
|
| 9999999980 |
\vec{\nabla} \cdot \vec{B} = 0
|
|
|
|
|
| 9999999981 |
\vec{\nabla} \cdot \vec{E} = \rho/\epsilon_0
|
|
|
|
|
| 9999999982 |
V = I R + Q/C + L \frac{\partial I}{\partial t}
|
|
|
|
|
| 9999999983 |
C = V A/d
|
|
|
|
|
| 9999999984 |
Q = C V
|
|
|
|
|
| 9999999985 |
V = I R
|
|
|
|
|
| 9999999986 |
\left[\right]\left[\right] = \left[\right]
|
|
|
|
|
| 9999999987 |
1 atmosphere = 101.325 Pascal
|
|
|
|
|
| 9999999988 |
1 atmosphere = 14.7 pounds/(inch^2)
|
|
|
|
|
| 9999999989 |
mass_{electron} = 511000 electronVolts/(q^2)
|
|
|
|
|
| 9999999990 |
Tesla = 10000*Gauss
|
|
|
|
|
| 9999999991 |
Pascal = Newton / (meter^2)
|
|
|
|
|
| 9999999992 |
Tesla = Newton / (Ampere*meter)
|
|
|
|
|
| 9999999993 |
Farad = Coulumb / Volt
|
|
|
|
|
| 9999999995 |
Volt = Joule / Coulumb
|
|
|
|
|
| 9999999996 |
Coulomb = Ampere / second
|
|
|
|
|
| 9999999997 |
Watt = Joule / second
|
|
|
|
|
| 9999999998 |
Joule = Newton*meter
|
|
|
|
|
| 9999999999 |
Newton = kilogram*meter/(second^2)
|
|
|
|
|
| 4961662865 |
x
|
|
|
|
|
| 9110536742 |
2 x
|
|
|
|
|
| 8483686863 |
\sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x)\right)
|
|
|
|
|
| 3470587782 |
\sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
|
|
|
|
|
| 8642992037 |
2
|
|
|
|
|
| 9894826550 |
2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \left(\exp(i x)+\exp(-i x)\right)
|
|
|
|
|
| 4257214632 |
\frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)
|
|
|
|
|
| 8699789241 |
2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)
|
|
|
|
|
| 9180861128 |
2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)
|
|
|
|
|
| 2405307372 |
\sin(2 x) = 2 \sin(x) \cos(x)
|
|
|
|
|
| 3268645065 |
x
|
|
|
|
|
| 9350663581 |
\pi
|
|
|
|
|
| 8332931442 |
\exp(i \pi) = \cos(\pi)+i \sin(\pi)
|
|
|
|
|
| 6885625907 |
\exp(i \pi) = -1 + i 0
|
|
|
|
|
| 3331824625 |
\exp(i \pi) = -1
|
|
|
|
|
| 4901237716 |
1
|
|
|
|
|
| 2501591100 |
\exp(i \pi) + 1 = 0
|
|
|
|
|
| 5136652623 |
E = KE + PE
|
|
mechanical energy is the sum of the potential plus kinetic energies |
|
|
| 7915321076 |
E, KE, PE
|
|
|
|
|
| 3192374582 |
E_2, KE_2, PE_2
|
|
|
|
|
| 7875206161 |
E_2 = KE_2 + PE_2
|
|
|
|
|
| 4229092186 |
E, KE, PE
|
|
|
|
|
| 1490821948 |
E_1, KE_1, PE_1
|
|
|
|
|
| 4303372136 |
E_1 = KE_1 + PE_1
|
|
|
|
|
| 5514556106 |
E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1)
|
|
|
|
|
| 8357234146 |
KE = \frac{1}{2} m v^2
|
|
kinetic energy |
Kinetic_energy
|
|
| 3658066792 |
KE_2, v_2
|
|
|
|
|
| 8082557839 |
KE, v
|
|
|
|
|
| 7676652285 |
KE_2 = \frac{1}{2} m v_2^2
|
|
|
|
|
| 2790304597 |
KE_1, v_1
|
|
|
|
|
| 9880327189 |
KE, v
|
|
|
|
|
| 4928007622 |
KE_1 = \frac{1}{2} m v_1^2
|
|
|
|
|
| 5733146966 |
KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right)
|
|
|
|
|
| 6554292307 |
t
|
|
|
|
|
| 4270680309 |
\frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t}
|
|
|
|
|
| 5781981178 |
x^2 - y^2 = (x+y)(x-y)
|
|
difference of squares |
Difference_of_two_squares
|
|
| 5946759357 |
v_2, v_1
|
|
|
|
|
| 6675701064 |
x, y
|
|
|
|
|
| 4648451961 |
v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1)
|
|
|
|
|
| 9356924046 |
\frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t}
|
|
|
|
|
| 9397152918 |
v = \frac{v_1 + v_2}{2}
|
|
average velocity |
|
|
| 2857430695 |
a = \frac{v_2 - v_1}{t}
|
|
acceleration |
|
|
| 7735737409 |
\frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t}
|
|
|
|
|
| 4784793837 |
\frac{KE_2 - KE_1}{t} = m v a
|
|
|
|
|
| 5345738321 |
F = m a
|
|
Newton's second law of motion |
Newton%27s_laws_of_motion#Newton's_second_law
|
|
| 2186083170 |
\frac{KE_2 - KE_1}{t} = v F
|
|
|
|
|
| 6715248283 |
PE = -F x
|
|
potential energy |
Potential_energy
|
|
| 3852078149 |
PE_2, x_2
|
|
|
|
|
| 7388921188 |
PE, x
|
|
|
|
|
| 2431507955 |
PE_2 = -F x_2
|
|
|
|
|
| 3412641898 |
PE_1, x_1
|
|
|
|
|
| 9937169749 |
PE, x
|
|
|
|
|
| 4669290568 |
PE_1 = -F x_1
|
|
|
|
|
| 7734996511 |
PE_2 - PE_1 = -F ( x_2 - x_1 )
|
|
|
|
|
| 2016063530 |
t
|
|
|
|
|
| 7267155233 |
\frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right)
|
|
|
|
|
| 9337785146 |
v = \frac{x_2 - x_1}{t}
|
|
average velocity |
|
|
| 4872970974 |
\frac{PE_2 - PE_1}{t} = -F v
|
|
|
|
|
| 2081689540 |
t
|
|
|
|
|
| 2770069250 |
\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t}
|
|
|
|
|
| 3591237106 |
\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v
|
|
|
|
|
| 1772416655 |
\frac{E_2 - E_1}{t} = v F - F v
|
|
|
|
|
| 1809909100 |
\frac{E_2 - E_1}{t} = 0
|
|
|
|
|
| 5778176146 |
t
|
|
|
|
|
| 3806977900 |
E_2 - E_1 = 0
|
|
|
|
|
| 5960438249 |
E_1
|
|
|
|
|
| 8558338742 |
E_2 = E_1
|
|
|
|
|
| 8945218208 |
\theta_{Brewster} + \theta_{refracted} = 90^{\circ}
|
|
|
based on figure 34-27 on page 824 in 2001_HRW
|
|
| 9025853427 |
\theta_{Brewster}
|
|
|
|
|
| 1310571337 |
\theta_{refracted} = 90^{\circ} - \theta_{Brewster}
|
|
|
|
|
| 6450985774 |
n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 )
|
|
Law of Refraction |
eq 34-44 on page 819 in 2001_HRW
|
|
| 1779497048 |
\theta_{Brewster}, \theta_{refraction}
|
|
|
|
|
| 7322344455 |
\theta_1, \theta_2
|
|
|
|
|
| 2575937347 |
n_1 \sin( \theta_{Brewster} ) = n_2 \sin( \theta_{refraction} )
|
|
|
|
|
| 7696214507 |
n_1 \sin( \theta_{Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{Brewster} )
|
|
|
|
|
| 8588429722 |
\sin( 90^{\circ} - x ) = \cos( x )
|
|
|
|
|
| 7375348852 |
\theta_{Brewster}
|
|
|
|
|
| 1512581563 |
x
|
|
|
|
|
| 6831637424 |
\sin( 90^{\circ} - \theta_{Brewster} ) = \cos( \theta_{Brewster} )
|
|
|
|
|
| 3061811650 |
n_1 \sin( \theta_{Brewster} ) = n_2 \cos( \theta_{Brewster} )
|
|
|
|
|
| 7857757625 |
n_1
|
|
|
|
|
| 9756089533 |
\sin( \theta_{Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{Brewster} )
|
|
|
|
|
| 5632428182 |
\cos( \theta_{Brewster} )
|
|
|
|
|
| 2768857871 |
\frac{\sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )} = \frac{n_2}{n_1}
|
|
|
|
|
| 4968680693 |
\tan( x ) = \frac{ \sin( x )}{\cos( x )}
|
|
|
|
|
| 7321695558 |
\theta_{Brewster}
|
|
|
|
|
| 9906920183 |
x
|
|
|
|
|
| 4501377629 |
\tan( \theta_{Brewster} ) = \frac{ \sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )}
|
|
|
|
|
| 3417126140 |
\tan( \theta_{Brewster} ) = \frac{ n_2 }{ n_1 }
|
|
|
|
|
| 5453995431 |
\arctan{ x }
|
|
|
|
|
| 6023986360 |
x
|
|
|
|
|
| 8495187962 |
\theta_{Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }
|
|
|
|
|
| 9040079362 |
f
|
|
|
|
|
| 9565166889 |
T
|
|
|
|
|
| 5426308937 |
v = \frac{d}{t}
|
|
|
|
|
| 7476820482 |
C
|
|
|
|
|
| 1277713901 |
d
|
|
|
|
|
| 6946088325 |
v = \frac{C}{t}
|
|
|
|
|
| 6785303857 |
C = 2 \pi r
|
|
|
|
|
| 4816195267 |
C, r
|
|
|
|
|
| 2113447367 |
C_{\rm{Earth\ orbit}}, r_{\rm{Earth\ orbit}}
|
|
|
|
|
| 6348260313 |
C_{\rm{Earth\ orbit}} = 2 \pi r_{\rm{Earth\ orbit}}
|
|
|
|
|
| 3847777611 |
C, v, t
|
|
|
|
|
| 6406487188 |
C_{\rm{Earth\ orbit}}, v_{\rm{Earth\ orbit}}, t_{\rm{Earth\ orbit}}
|
|
|
|
|
| 3046191961 |
v_{\rm{Earth\ orbit}} = \frac{C_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
|
|
|
|
|
| 3080027960 |
v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
|
|
|
|
|
| 7175416299 |
t_{\rm{Earth\ orbit}} = 1 {\rm{year}}
|
|
|
|
|
| 3219318145 |
\frac{365 {\rm days}}{1 {\rm year}} \frac{24 {\rm hours}}{1 {\rm day}} \frac{60 {\rm minutes}}{1 {\rm hour}} \frac{60 {\rm seconds}}{1 {\rm minute}}
|
|
|
|
|
| 8721295221 |
t_{\rm{Earth\ orbit}} = 3.16 10^7 {\rm{seconds}}
|
|
|
|
|
| 4593428198 |
v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{3.16\ 10^7 {\rm seconds}}
|
|
|
|
|
| 3472836147 |
r_{\rm{Earth\ orbit}} = 1.496\ 10^8 {\rm km}
|
|
|
|
|
| 6998364753 |
v_{\rm{Earth\ orbit}} = \frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}}
|
|
|
|
|
| 4180845508 |
v_{\rm{Earth\ orbit}} = 29.8 \frac{{\rm km}}{{\rm sec}}
|
|
|
|
|
| 4662369843 |
x' = \gamma (x - v t)
|
|
|
|
|
| 2983053062 |
x = \gamma (x' + v t')
|
|
|
|
|
| 3426941928 |
x = \gamma ( \gamma (x - v t) + v t' )
|
|
|
|
|
| 2096918413 |
x = \gamma ( \gamma x - \gamma v t + v t' )
|
|
|
|
|
| 7741202861 |
x = \gamma^2 x - \gamma^2 v t + \gamma v t'
|
|
|
|
|
| 7337056406 |
\gamma^2 x
|
|
|
|
|
| 4139999399 |
x - \gamma^2 x = - \gamma^2 v t + \gamma v t'
|
|
|
|
|
| 3495403335 |
x
|
|
|
|
|
| 9031609275 |
x (1 - \gamma^2 ) = - \gamma^2 v t + \gamma v t'
|
|
|
|
|
| 8014566709 |
\gamma^2 v t
|
|
|
|
|
| 9409776983 |
x (1 - \gamma^2 ) + \gamma^2 v t = \gamma v t'
|
|
|
|
|
| 2226340358 |
\gamma v
|
|
|
|
|
| 6417705759 |
\frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = \gamma v t'
|
|
|
|
|
| 8730201316 |
\frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t = t'
|
|
|
first term was multiplied by \gamma/\gamma
|
|
| 1974334644 |
\frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = t'
|
|
|
|
|
| 5148266645 |
t' = \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t
|
|
|
|
|
| 4287102261 |
x^2 + y^2 + z^2 = c^2 t^2
|
|
|
describes a spherical wavefront
|
|
| 1201689765 |
x'^2 + y'^2 + z'^2 = c^2 t'^2
|
|
|
describes a spherical wavefront for an observer in a moving frame of reference
|
|
| 7057864873 |
y' = y
|
|
|
frame of reference is moving only along x direction
|
|
| 8515803375 |
z' = z
|
|
|
frame of reference is moving only along x direction
|
|
| 6153463771 |
3
|
|
|
|
|
| 4746576736 |
asdf
|
|
|
|
|
| 1027270623 |
minga
|
|
|
|
|
| 6101803533 |
ming
|
|
|
|
|
| 9231495607 |
a = b
|
|
|
|
|
| 8664264194 |
b + 2
|
|
|
|
|
| 9805063945 |
\gamma^2 (x - v t)^2 + y^2 + z^2 = c^2 \gamma^2 \left( t + \frac{ 1 - \gamma^2 }{ \gamma^2 } \frac{x}{v} \right)^2
|
|
|
|
|
| 4057686137 |
C \to C_{\rm{Earth\ orbit}}
|
|
|
|
|
| 2346150725 |
r \to r_{\rm{Earth\ orbit}}
|
|
|
|
|
| 9753878784 |
v \to v_{\rm{Earth\ orbit}}
|
|
|
|
|
| 8135396036 |
t \to t_{\rm{Earth\ orbit}}
|
|
|
|
|
| 2773628333 |
\theta_1 \to \theta_{Brewster}
|
|
|
|
|
| 7154592211 |
\theta_2 \to \theta_{\rm refracted}
|
|
|
|
|
| 3749492596 |
E \to E_1
|
|
|
|
|
| 1258245373 |
E \to E_2
|
|
|
|
|
| 4147101187 |
KE \to KE_1
|
|
|
|
|
| 3809726424 |
PE \to PE_1
|
|
|
|
|
| 6383056612 |
KE \to KE_2
|
|
|
|
|
| 5075406409 |
PE \to PE_2
|
|
|
|
|
| 5398681503 |
v \to v_1
|
|
|
|
|
| 9305761407 |
v \to v_2
|
|
|
|
|
| 4218009993 |
x \to x_1
|
|
|
|
|
| 4188639044 |
x \to x_2
|
|
|
|
|
| 8173074178 |
x \to v_2
|
|
|
|
|
| 1025759423 |
y \to v_1
|
|
|
|
|
| 1935543849 |
\gamma^2 x^2 - \gamma^2 2 x v t + \gamma^2 v^2 t^2 + y^2 + z^2 = c^2 \gamma^2 \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x^2}{\gamma^2} + c^2 \gamma^2 2 t \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x}{\gamma} + c^2 \gamma^2 t^2
|
|
|
|
|
| 1586866563 |
\left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right) = t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)
|
|
|
|
|
| 3182633789 |
\gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1
|
|
|
|
|
| 1916173354 |
-\gamma^2 v^2 + c^2 \gamma^2 = c^2
|
|
|
|
|
| 2076171250 |
-\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} = 0
|
|
|
|
|
| 5284610349 |
\gamma^2
|
|
|
|
|
| 2417941373 |
- c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 - \gamma^2
|
|
|
|
|
| 5787469164 |
1 - \gamma^2
|
|
|
|
|
| 1639827492 |
- c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2} = 1
|
|
|
|
|
| 5669500954 |
v^2 \gamma^2
|
|
|
|
|
| 5763749235 |
-c^2 + c^2 \gamma^2 = v^2 \gamma^2
|
|
|
|
|
| 6408214498 |
c^2
|
|
|
|
|
| 2999795755 |
c^2 \gamma^2 = v^2 \gamma^2 + c^2
|
|
|
|
|
| 3412946408 |
v^2 \gamma^2
|
|
|
|
|
| 2542420160 |
c^2 \gamma^2 - v^2 \gamma^2 = c^2
|
|
|
|
|
| 7743841045 |
\gamma^2
|
|
|
|
|
| 7513513483 |
\gamma^2 (c^2 - v^2) = c^2
|
|
|
|
|
| 8571466509 |
c^2 - \gamma^2
|
|
|
|
|
| 7906112355 |
\gamma^2 = \frac{c^2}{c^2 - \gamma^2}
|
|
|
|
|
| 3060139639 |
1/2
|
|
|
|
|
| 1528310784 |
\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
|
|
|
|
|
| 8360117126 |
\gamma = \frac{-1}{\sqrt{1-\frac{v^2}{c^2}}}
|
|
|
not a physically valid result in this context
|
|
| 3366703541 |
a = \frac{v - v_0}{t}
|
|
|
acceleration is the average change in speed over a duration
|
|
| 7083390553 |
t
|
|
|
|
|
| 4748157455 |
a t = v - v_0
|
|
|
|
|
| 6417359412 |
v_0
|
|
|
|
|
| 4798787814 |
a t + v_0 = v
|
|
|
|
|
| 3462972452 |
v = v_0 + a t
|
|
|
|
|
| 3411994811 |
v_{\rm ave} = \frac{d}{t}
|
|
|
|
|
| 6175547907 |
v_{\rm ave} = \frac{v + v_0}{2}
|
|
|
|
|
| 9897284307 |
\frac{d}{t} = \frac{v + v_0}{2}
|
|
|
|
|
| 8865085668 |
t
|
|
|
|
|
| 8706092970 |
d = \left(\frac{v + v_0}{2}\right)t
|
|
|
|
|
| 7011114072 |
d = \frac{(v_0 + a t) + v_0}{2} t
|
|
|
|
|
| 1265150401 |
d = \frac{2 v_0 + a t}{2} t
|
|
|
|
|
| 9658195023 |
d = v_0 t + \frac{1}{2} a t^2
|
|
|
|
|
| 5799753649 |
2
|
|
|
|
|
| 7215099603 |
v^2 = v_0^2 + 2 a t v_0 + a^2 t^2
|
|
|
|
|
| 5144263777 |
v^2 = v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right)
|
|
|
|
|
| 7939765107 |
v^2 = v_0^2 + 2 a d
|
|
|
|
|
| 6729698807 |
v_0
|
|
|
|
|
| 9759901995 |
v - v_0 = a t
|
|
|
|
|
| 2242144313 |
a
|
|
|
|
|
| 1967582749 |
t = \frac{v - v_0}{a}
|
|
|
|
|
| 5733721198 |
d = \frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right)
|
|
|
|
|
| 5611024898 |
d = \frac{1}{2 a} (v^2 - v_0^2)
|
|
|
|
|
| 5542390646 |
2 a
|
|
|
|
|
| 8269198922 |
2 a d = v^2 - v_0^2
|
|
|
|
|
| 9070454719 |
v_0^2
|
|
|
|
|
| 4948763856 |
2 a d + v_0^2 = v^2
|
|
|
|
|
| 9645178657 |
a t
|
|
|
|
|
| 6457044853 |
v - a t = v_0
|
|
|
|
|
| 1259826355 |
d = (v - a t) t + \frac{1}{2} a t^2
|
|
|
|
|
| 4580545876 |
d = v t - a t^2 + \frac{1}{2} a t^2
|
|
|
|
|
| 6421241247 |
d = v t - \frac{1}{2} a t^2
|
|
|
|
|
| 4182362050 |
Z = |Z| \exp( i \theta )
|
|
|
Z \in \mathbb{C}
|
|
| 1928085940 |
Z^* = |Z| \exp( -i \theta )
|
|
|
|
|
| 9191880568 |
Z Z^* = |Z| |Z| \exp( -i \theta ) \exp( i \theta )
|
|
|
|
|
| 3350830826 |
Z Z^* = |Z|^2
|
|
|
|
|
| 8396997949 |
I = | A + B |^2
|
|
|
intensity of two waves traveling opposite directions on same path
|
|
| 4437214608 |
Z
|
|
|
|
|
| 5623794884 |
A + B
|
|
|
|
|
| 2236639474 |
(A + B)(A + B)^* = |A + B|^2
|
|
|
|
|
| 1020854560 |
I = (A + B)(A + B)^*
|
|
|
|
|
| 6306552185 |
I = (A + B)(A^* + B^*)
|
|
|
|
|
| 8065128065 |
I = A A^* + B B^* + A B^* + B A^*
|
|
|
|
|
| 9761485403 |
Z
|
|
|
|
|
| 8710504862 |
A
|
|
|
|
|
| 4075539836 |
A A^* = |A|^2
|
|
|
|
|
| 6529120965 |
B
|
|
|
|
|
| 1511199318 |
Z
|
|
|
|
|
| 7107090465 |
B B^* = |B|^2
|
|
|
|
|
| 5125940051 |
I = |A|^2 + B B^* + A B^* + B A^*
|
|
|
|
|
| 1525861537 |
I = |A|^2 + |B|^2 + A B^* + B A^*
|
|
|
|
|
| 1742775076 |
Z \to B
|
|
|
|
|
| 7607271250 |
\theta \to \phi
|
|
|
|
|
| 4192519596 |
B = |B| \exp(i \phi)
|
|
|
|
|
| 2064205392 |
A
|
|
|
|
|
| 1894894315 |
Z
|
|
|
|
|
| 1357848476 |
A = |A| \exp(i \theta)
|
|
|
|
|
| 6555185548 |
A^* = |A| \exp(-i \theta)
|
|
|
|
|
| 4504256452 |
B^* = |B| \exp(-i \phi)
|
|
|
|
|
| 7621705408 |
I = |A|^2 + |B|^2 + |A| |B| \exp(-i \theta) \exp(i \phi) + |A| |B| \exp(i \theta) \exp(-i \phi)
|
|
|
|
|
| 3085575328 |
I = |A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi))
|
|
|
|
|
| 4935235303 |
x
|
|
|
|
|
| 2293352649 |
\theta - \phi
|
|
|
|
|
| 3660957533 |
\cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
|
|
|
|
|
| 3967985562 |
2
|
|
|
|
|
| 2700934933 |
2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
|
|
|
|
|
| 8497631728 |
I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )
|
|
|
|
|
| 6774684564 |
\theta = \phi
|
|
|
for coherent waves
|
|
| 2099546947 |
I = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )
|
|
|
|
|
| 2719691582 |
|A| = |B|
|
|
|
in a loop
|
|
| 3257276368 |
I = |A|^2 + |A|^2 + |A| |A| 2
|
|
|
|
|
| 8283354808 |
I_{\rm coherent} = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )
|
|
|
|
|
| 8046208134 |
I_{\rm coherent} = |A|^2 + |A|^2 + |A| |A| 2
|
|
|
|
|
| 1172039918 |
I_{\rm coherent} = 4 |A|^2
|
|
|
|
|
| 8602221482 |
\langle \cos(\theta - \phi) \rangle = 0
|
|
|
incoherent light source
|
|
| 6240206408 |
I_{\rm incoherent} = |A|^2 + |B|^2
|
|
|
|
|
| 6529793063 |
I_{\rm incoherent} = |A|^2 + |A|^2
|
|
|
|
|
| 3060393541 |
I_{\rm incoherent} = 2|A|^2
|
|
|
|
|
| 6556875579 |
\frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2
|
|
|
|
|
| 8750500372 |
f
|
|
|
|
|
| 7543102525 |
g
|
|
|
|
|
| 8267133829 |
a = b
|
|
|
|
|
| 7968822469 |
c = d
|
|
|
|
|
| 3169580383 |
\vec{a} = \frac{d\vec{v}}{dt}
|
|
|
acceleration is the change in speed over a duration
|
|
| 8602512487 |
\vec{a} = a_x \hat{x} + a_y \hat{y}
|
|
|
decompose acceleration into two components
|
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| 4158986868 |
a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt}
|
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|
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| 5349866551 |
\vec{v} = v_x \hat{x} + v_y \hat{y}
|
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|
|
|
| 7729413831 |
a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right)
|
|
|
|
|
| 1819663717 |
a_x = \frac{d}{dt} v_x
|
|
|
|
|
| 8228733125 |
a_y = \frac{d}{dt} v_y
|
|
|
|
|
| 9707028061 |
a_x = 0
|
|
|
|
|
| 2741489181 |
a_y = -g
|
|
|
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| 3270039798 |
2
|
|
|
|
|
| 8880467139 |
2
|
|
|
|
|
| 8750379055 |
0 = \frac{d}{dt} v_x
|
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|
|
|
| 1977955751 |
-g = \frac{d}{dt} v_y
|
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|
|
| 6672141531 |
dt
|
|
|
|
|
| 1702349646 |
-g dt = d v_y
|
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|
|
|
| 8584698994 |
-g \int dt = \int d v_y
|
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|
|
|
| 9973952056 |
-g t = v_y - v_{0, y}
|
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|
|
| 4167526462 |
v_{0, y}
|
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|
|
|
| 6572039835 |
-g t + v_{0, y} = v_y
|
|
|
|
|
| 7252338326 |
v_y = \frac{dy}{dt}
|
|
|
|
|
| 6204539227 |
-g t + v_{0, y} = \frac{dy}{dt}
|
|
|
|
|
| 1614343171 |
dt
|
|
|
|
|
| 8145337879 |
-g t dt + v_{0, y} dt = dy
|
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|
|
|
| 8808860551 |
-g \int t dt + v_{0, y} \int dt = \int dy
|
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|
|
| 2858549874 |
- \frac{1}{2} g t^2 + v_{0, y} t = y - y_0
|
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|
|
| 6098638221 |
y_0
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|
|
| 2461349007 |
- \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y
|
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|
|
|
| 8717193282 |
dt
|
|
|
|
|
| 1166310428 |
0 dt = d v_x
|
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|
|
|
| 2366691988 |
\int 0 dt = \int d v_x
|
|
|
|
|
| 1676472948 |
0 = v_x - v_{0, x}
|
|
|
|
|
| 1439089569 |
v_{0, x}
|
|
|
|
|
| 6134836751 |
v_{0, x} = v_x
|
|
|
|
|
| 8460820419 |
v_x = \frac{dx}{dt}
|
|
|
|
|
| 7455581657 |
v_{0, x} = \frac{dx}{dt}
|
|
|
|
|
| 8607458157 |
dt
|
|
|
|
|
| 1963253044 |
v_{0, x} dt = dx
|
|
|
|
|
| 3676159007 |
v_{0, x} \int dt = \int dx
|
|
|
|
|
| 9882526611 |
v_{0, x} t = x - x_0
|
|
|
|
|
| 3182907803 |
x_0
|
|
|
|
|
| 8486706976 |
v_{0, x} t + x_0 = x
|
|
|
|
|
| 1306360899 |
x = v_{0, x} t + x_0
|
|
|
|
|
| 7049769409 |
2
|
|
|
|
|
| 9341391925 |
\vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y}
|
|
|
|
|
| 6410818363 |
\theta
|
|
|
|
|
| 7391837535 |
\cos(\theta) = \frac{v_{0, x}}{v_0}
|
|
|
|
|
| 7376526845 |
\sin(\theta) = \frac{v_{0, y}}{v_0}
|
|
|
|
|
| 5868731041 |
v_0
|
|
|
|
|
| 6083821265 |
v_0 \cos(\theta) = v_{0, x}
|
|
|
|
|
| 5438722682 |
x = v_0 t \cos(\theta) + x_0
|
|
|
|
|
| 5620558729 |
v_0
|
|
|
|
|
| 8949329361 |
v_0 \sin(\theta) = v_{0, y}
|
|
|
|
|
| 1405465835 |
y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0
|
|
|
|
|
| 9862900242 |
y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0
|
|
|
|
|
| 6050070428 |
v_{0, x}
|
|
|
|
|
| 3274926090 |
t = \frac{x - x_0}{v_{0, x}}
|
|
|
|
|
| 7354529102 |
y = - \frac{1}{2} g \left( \frac{x - x_0}{v_{0, x}} \right)^2 + v_{0, y} \frac{x - x_0}{v_{0, x}} + y_0
|
|
|
|
|
| 8406170337 |
y \to y_f
|
|
|
|
|
| 2403773761 |
t \to t_f
|
|
|
|
|
| 5379546684 |
y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
|
|
|
|
|
| 5373931751 |
t = t_f
|
|
|
|
|
| 9112191201 |
y_f = 0
|
|
|
|
|
| 8198310977 |
0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
|
|
|
|
|
| 1650441634 |
y_0 = 0
|
|
|
define coordinate system such that initial height is at origin
|
|
| 1087417579 |
0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)
|
|
|
|
|
| 4829590294 |
t_f
|
|
|
|
|
| 2086924031 |
0 = - \frac{1}{2} g t_f + v_0 \sin(\theta)
|
|
|
|
|
| 6974054946 |
\frac{1}{2} g t_f
|
|
|
|
|
| 1191796961 |
\frac{1}{2} g t_f = v_0 \sin(\theta)
|
|
|
|
|
| 2510804451 |
2/g
|
|
|
|
|
| 4778077984 |
t_f = \frac{2 v_0 \sin(\theta)}{g}
|
|
|
|
|
| 3273630811 |
x \to x_f
|
|
|
|
|
| 6732786762 |
t \to t_f
|
|
|
|
|
| 3485125659 |
x_f = v_0 t_f \cos(\theta) + x_0
|
|
|
|
|
| 4370074654 |
t = t_f
|
|
|
|
|
| 2378095808 |
x_f = x_0 + d
|
|
|
|
|
| 4268085801 |
x_0 + d = v_0 t_f \cos(\theta) + x_0
|
|
|
|
|
| 8072682558 |
x_0
|
|
|
|
|
| 7233558441 |
d = v_0 t_f \cos(\theta)
|
|
|
|
|
| 2297105551 |
d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)
|
|
|
|
|
| 7587034465 |
\theta
|
|
|
|
|
| 7214442790 |
x
|
|
|
|
|
| 2519058903 |
\sin(2 \theta) = 2 \sin(\theta) \cos(\theta)
|
|
|
|
|
| 8922441655 |
d = \frac{v_0^2}{g} \sin(2 \theta)
|
|
|
|
|
| 5667870149 |
\theta
|
|
|
|
|
| 1541916015 |
\theta = \frac{\pi}{4}
|
|
|
|
|
| 3607070319 |
d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)
|
|
|
|
|
| 5353282496 |
d = \frac{v_0^2}{g}
|
|
|
|
|
| 0000040490 |
a^2
|
|
|
|
|
| 0000999900 |
b/(2a)
|
|
|
|
|
| 0001030901 |
\cos(x)
|
|
|
|
|
| 0001111111 |
(\sin(x))^2
|
|
|
|
|
| 0001209482 |
2 \pi
|
|
|
|
|
| 0001304952 |
\hbar
|
|
|
|
|
| 0001334112 |
W
|
|
|
|
|
| 0001921933 |
2 i
|
|
|
|
|
| 0002239424 |
2
|
|
|
|
|
| 0002338514 |
\vec{p}_{2}
|
|
|
|
|
| 0002342425 |
m/m
|
|
|
|
|
| 0002367209 |
1/2
|
|
|
|
|
| 0002393922 |
x
|
|
|
|
|
| 0002424922 |
a
|
|
|
|
|
| 0002436656 |
i \hbar
|
|
|
|
|
| 0002449291 |
b/(2a)
|
|
|
|
|
| 0002838490 |
b/(2a)
|
|
|
|
|
| 0002919191 |
\sin(-x)
|
|
|
|
|
| 0002929944 |
1/2
|
|
|
|
|
| 0002940021 |
2 \pi
|
|
|
|
|
| 0003232242 |
t
|
|
|
|
|
| 0003413423 |
\cos(-x)
|
|
|
|
|
| 0003747849 |
-1
|
|
|
|
|
| 0003838111 |
2
|
|
|
|
|
| 0003919391 |
x
|
|
|
|
|
| 0003949052 |
-x
|
|
|
|
|
| 0003949921 |
\hbar
|
|
|
|
|
| 0003954314 |
dx
|
|
|
|
|
| 0003981813 |
-\sin(x)
|
|
|
|
|
| 0004089571 |
2x
|
|
|
|
|
| 0004264724 |
y
|
|
|
|
|
| 0004307451 |
(b/(2a))^2
|
|
|
|
|
| 0004582412 |
x
|
|
|
|
|
| 0004829194 |
2
|
|
|
|
|
| 0004831494 |
a
|
|
|
|
|
| 0004849392 |
x
|
|
|
|
|
| 0004858592 |
h
|
|
|
|
|
| 0004934845 |
x
|
|
|
|
|
| 0004948585 |
a
|
|
|
|
|
| 0005395034 |
a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
|
|
|
|
|
| 0005626421 |
t
|
|
|
|
|
| 0005749291 |
f
|
|
|
|
|
| 0005938585 |
\frac{-\hbar^2}{2m}
|
|
|
|
|
| 0006466214 |
(\sin(x))^2
|
|
|
|
|
| 0006544644 |
t
|
|
|
|
|
| 0006563727 |
x
|
|
|
|
|
| 0006644853 |
c/a
|
|
|
|
|
| 0006656532 |
e
|
|
|
|
|
| 0007471778 |
2(\sin(x))^2
|
|
|
|
|
| 0007563791 |
i
|
|
|
|
|
| 0007636749 |
x
|
|
|
|
|
| 0007894942 |
(\sin(x))^2
|
|
|
|
|
| 0008837284 |
T
|
|
|
|
|
| 0008842811 |
\cos(2x)
|
|
|
|
|
| 0009458842 |
\psi(x)
|
|
|
|
|
| 0009484724 |
\frac{n \pi}{W}x
|
|
|
|
|
| 0009485857 |
a^2\frac{2}{W}
|
|
|
|
|
| 0009485858 |
\frac{2n\pi}{W}
|
|
|
|
|
| 0009492929 |
v du
|
|
|
|
|
| 0009587738 |
\psi
|
|
|
|
|
| 0009594995 |
1/2
|
|
|
|
|
| 0009877781 |
y
|
|
|
|
|
| 0203024440 |
1 = \int_0^W a \sin(\frac{n \pi}{W} x) \psi(x)^* dx
|
|
|
|
|
| 0404050504 |
\lambda = \frac{v}{f}
|
|
|
|
|
| 0439492440 |
\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)\right|_0^W
|
|
|
|
|
| 0934990943 |
k = \frac{2 \pi}{v T}
|
|
|
|
|
| 0948572140 |
\int \cos(a x) dx = \frac{1}{a}\sin(a x)
|
|
|
|
|
| 1010393913 |
\langle \psi| \hat{A}^+ |\psi \rangle = \langle a \rangle^*
|
|
|
|
|
| 1010393944 |
x = \langle\psi_{\alpha}| a_{\beta} |\psi_{\beta} \rangle
|
|
|
|
|
| 1010923823 |
k W = n \pi
|
|
|
|
|
| 1020010291 |
0 = a \sin(k W)
|
|
|
|
|
| 1020394900 |
p = h/\lambda
|
|
|
|
|
| 1020394902 |
E = h f
|
|
|
|
|
| 1029039903 |
p = m v
|
|
|
|
|
| 1029039904 |
p^2 = m^2 v^2
|
|
|
|
|
| 1158485859 |
\frac{-\hbar^2}{2m} \nabla^2 = {\cal H}
|
|
|
|
|
| 1202310110 |
\frac{1}{a^2} = \int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
|
|
|
|
|
| 1202312210 |
\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
|
|
|
|
|
| 1203938249 |
a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle = a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
|
|
|
|
|
| 1248277773 |
\cos(2x) = 1-2(\sin(x))^2
|
|
|
|
|
| 1293913110 |
0 = b
|
|
|
|
|
| 1293923844 |
\lambda = v T
|
|
|
|
|
| 1314464131 |
\vec{\nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
|
|
|
|
|
| 1314864131 |
\vec{\nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E}
|
|
|
|
|
| 1395858355 |
x = \langle \psi_{\alpha}| a_{\alpha} |\psi_{\beta}\rangle
|
|
|
|
|
| 1492811142 |
f = x - d
|
|
|
|
|
| 1492842000 |
\nabla \vec{x} = f(y)
|
|
|
|
|
| 1636453295 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = - \nabla^2 \vec{E}
|
|
|
|
|
| 1638282134 |
\vec{p}_{before} = \vec{p}_{after}
|
|
|
|
|
| 1648958381 |
\nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{\nabla}\psi(\vec{r},t)\right)
|
|
|
|
|
| 1857710291 |
0 = a \sin(n \pi)
|
|
|
|
|
| 1858578388 |
\nabla^2 E(\vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 1858772113 |
k = \frac{n \pi}{W}
|
|
|
|
|
| 1934748140 |
\int |\psi(x)|^2 dx = 1
|
|
|
|
|
| 2029293929 |
\nabla^2 E(\vec{r})\exp(i \omega t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 2103023049 |
\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right)
|
|
|
|
|
| 2113211456 |
f = 1/T
|
|
|
|
|
| 2123139121 |
-\exp(-i x) = -\cos(x)+i \sin(x)
|
|
|
|
|
| 2131616531 |
T f = 1
|
|
|
|
|
| 2258485859 |
{\cal H} \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
|
|
|
|
|
| 2394240499 |
x = a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle
|
|
|
|
|
| 2394853829 |
\exp(-i x) = \cos(-x)+i \sin(-x)
|
|
|
|
|
| 2394935831 |
( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
|
|
|
|
|
| 2394935835 |
\left(\langle\psi| \hat{A} |\psi\rangle \right)^+ = \left(\langle a \rangle\right)^+
|
|
|
|
|
| 2395958385 |
\nabla^2 \psi\left(\vec{r},t)\right) = \frac{-p^2}{\hbar} \psi(\vec{r},t)
|
|
|
|
|
| 2404934990 |
\langle x^2\rangle -2\langle x \rangle\langle x \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 2494533900 |
\langle x^2\rangle -\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 2569154141 |
\vec{\nabla} \times \frac{\partial}{\partial t}\vec{H} = \epsilon_0 \frac{\partial^2 }{\partial t^2}\vec{E}
|
|
|
|
|
| 2648958382 |
\nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p}\psi(\vec{r},t)\right)
|
|
|
|
|
| 2848934890 |
\langle a \rangle^* = \langle a \rangle
|
|
|
|
|
| 2944838499 |
\psi(x) = a \sin(\frac{n \pi}{W} x)
|
|
|
|
|
| 3121234211 |
\frac{k}{2\pi} = \lambda
|
|
|
|
|
| 3121234212 |
p = \frac{h k}{2\pi}
|
|
|
|
|
| 3121513111 |
k = \frac{2 \pi}{\lambda}
|
|
|
|
|
| 3131111133 |
T = 1 / f
|
|
|
|
|
| 3131211131 |
\omega = 2 \pi f
|
|
|
|
|
| 3132131132 |
\omega = \frac{2\pi}{T}
|
|
|
|
|
| 3147472131 |
\frac{\omega}{2 \pi} = f
|
|
|
|
|
| 3285732911 |
(\cos(x))^2 = 1-(\sin(x))^2
|
|
|
|
|
| 3485475729 |
\nabla^2 E(\vec{r}) = - \frac{\omega^2}{c^2} E(\vec{r})
|
|
|
|
|
| 3585845894 |
\langle \left(x-\langle x \rangle\right)^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 3829492824 |
\frac{1}{2}\left(\exp(i x)+\exp(-i x)\right) = \cos(x)
|
|
|
|
|
| 3924948349 |
a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
|
|
|
|
|
| 3942849294 |
\exp(i x)-\exp(-i x) = 2 i \sin(x)
|
|
|
|
|
| 3943939590 |
x = a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle
|
|
|
|
|
| 3947269979 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
|
|
|
|
|
| 3948571256 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi(\vec{r},t)
|
|
|
|
|
| 3948572230 |
\vec{\nabla}\psi(\vec{r},t) = \psi_0 \vec{\nabla}\exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
|
|
|
|
|
| 3948574224 |
\psi(\vec{r},t) = \psi_0 \exp\left(i\left(\vec{k}\cdot\vec{r} - \omega t \right) \right)
|
|
|
|
|
| 3948574226 |
\psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right)
|
|
|
|
|
| 3948574228 |
\psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
|
|
|
|
|
| 3948574230 |
\psi(\vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
|
|
|
|
|
| 3948574233 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
|
|
|
|
|
| 3948574235 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
|
|
|
|
|
| 3951205425 |
\vec{p}_{after} = \vec{p}_{1}
|
|
|
|
|
| 4147472132 |
E = \frac{h \omega}{2 \pi}
|
|
|
|
|
| 4298359835 |
E = \frac{1}{2}m v^2
|
|
|
|
|
| 4298359845 |
E = \frac{1}{2m}m^2 v^2
|
|
|
|
|
| 4298359851 |
E = \frac{p^2}{2m}
|
|
|
|
|
| 4341171256 |
i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{p^2}{2 m} \psi(\vec{r},t)
|
|
|
|
|
| 4348571256 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi(\vec{r},t)
|
|
|
|
|
| 4394958389 |
\vec{\nabla}\cdot \left(\vec{\nabla}\psi(\vec{r},t)\right) = \frac{i}{\hbar} \vec{\nabla}\cdot\left( \vec{p} \psi(\vec{r},t)\right)
|
|
|
|
|
| 4585828572 |
\epsilon_0 \mu_0 = \frac{1}{c^2}
|
|
|
|
|
| 4585932229 |
\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
|
|
|
|
|
| 4598294821 |
\exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
|
|
|
|
|
| 4638429483 |
\exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))
|
|
|
|
|
| 4742644828 |
\exp(i x)+\exp(-i x) = 2 \cos(x)
|
|
|
|
|
| 4827492911 |
\cos(2x)+(\sin(x))^2 = 1-(\sin(x))^2
|
|
|
|
|
| 4838429483 |
\exp(2 i x) = \cos(2 x)+i \sin(2 x)
|
|
|
|
|
| 4843995999 |
\frac{1}{2 i}\left(\exp(i x)-\exp(-i x)\right) = \sin(x)
|
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|
|
| 4857472413 |
1 = \int \psi(x)\psi(x)^* dx
|
|
|
|
|
| 4857475848 |
\frac{1}{a^2} = \frac{W}{2}
|
|
|
|
|
| 4858429483 |
\exp(i x)\exp(i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))
|
|
|
|
|
| 4923339482 |
i x = \log(y)
|
|
|
|
|
| 4928239482 |
\log(y) = i x
|
|
|
|
|
| 4928923942 |
a = b
|
|
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|
|
| 4929218492 |
a+b = c
|
|
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|
|
| 4929482992 |
b = 2
|
|
|
|
|
| 4938429482 |
\exp(-i x) = \cos(x)+i \sin(-x)
|
|
|
|
|
| 4938429483 |
\exp(i x) = \cos(x)+i \sin(x)
|
|
|
|
|
| 4938429484 |
\exp(-i x) = \cos(x)-i \sin(x)
|
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|
|
|
| 4943571230 |
\vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
|
|
|
|
|
| 4948934890 |
\langle \psi| \hat{A} |\psi\rangle = \langle a \rangle^*
|
|
|
|
|
| 4949359835 |
\langle x^2\rangle -2\langle x^2 \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 4954839242 |
\cos(2x)+i\sin(2x) = (\cos(x)+i \sin(x))(\cos(x)+i \sin(x))
|
|
|
|
|
| 4978429483 |
\exp(-i x) = \cos(-x)+i \sin(-x)
|
|
|
|
|
| 4984892984 |
a+2 = c
|
|
|
|
|
| 4985825552 |
\nabla^2 E(\vec{r})\exp(i \omega t) = i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 5530148480 |
\vec{p}_{1}-\vec{p}_{2} = \vec{p}_{electron}
|
|
|
|
|
| 5727578862 |
\frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)
|
|
|
|
|
| 5832984291 |
(\sin(x))^2 + (\cos(x))^2 = 1
|
|
|
|
|
| 5857434758 |
\int a dx = a x
|
|
|
|
|
| 5868688585 |
\frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = \frac{p^2}{2m} \psi(\vec{r},t)
|
|
|
|
|
| 5900595848 |
k = \frac{\omega}{v}
|
|
|
|
|
| 5928285821 |
x^2 + 2x(b/(2a)) + (b/(2a))^2 = (x+(b/(2a)))^2
|
|
|
|
|
| 5928292841 |
x^2 + (b/a)x + (b/(2a))^2 = -c/a + (b/(2a))^2
|
|
|
|
|
| 5938459282 |
x^2 + (b/a)x = -c/a
|
|
|
|
|
| 5958392859 |
x^2 + (b/a)x+(c/a) = 0
|
|
|
|
|
| 5959282914 |
x^2 + x(b/a) + (b/(2a))^2 = (x+(b/(2a)))^2
|
|
|
|
|
| 5982958248 |
x = -\sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
|
|
|
|
|
| 5982958249 |
x+(b/(2a)) = -\sqrt{(b/(2a))^2 - (c/a)}
|
|
|
|
|
| 5985371230 |
\vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi(\vec{r},t)
|
|
|
|
|
| 6742123016 |
\vec{p}_{electron}\cdot\vec{p}_{electron} = (\vec{p}_{1}\cdot\vec{p}_{1})+(\vec{p}_{2}\cdot\vec{p}_{2})-2(\vec{p}_{1}\cdot\vec{p}_{2})
|
|
|
|
|
| 7466829492 |
\vec{\nabla} \cdot \vec{E} = 0
|
|
|
|
|
| 7564894985 |
\int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)
|
|
|
|
|
| 7572664728 |
\cos(2x)+2(\sin(x))^2 = 1
|
|
|
|
|
| 7575738420 |
\left(\sin\left(\frac{n \pi}{W}x\right)\right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2}
|
|
|
|
|
| 7575859295 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859300 |
\epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{\nabla} \times \vec{E} )_k = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859302 |
\epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859304 |
\epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h}
|
|
|
|
|
| 7575859306 |
\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859308 |
\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right) = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859310 |
\hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859312 |
\vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7917051060 |
\vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2}
|
|
|
|
|
| 8139187332 |
\vec{p}_{1} = \vec{p}_{2}+\vec{p}_{electron}
|
|
|
|
|
| 8257621077 |
\vec{p}_{before} = \vec{p}_{1}
|
|
|
|
|
| 8311458118 |
\vec{p}_{after} = \vec{p}_{2}+\vec{p}_{electron}
|
|
|
|
|
| 8399484849 |
\langle x^2 -2x\langle x \rangle+\langle x \rangle^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 8484544728 |
-a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x)
|
|
|
|
|
| 8485757728 |
a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(kx)
|
|
|
|
|
| 8485867742 |
\frac{2}{W} = a^2
|
|
|
|
|
| 8489593958 |
d(u v) = u dv + v du
|
|
|
|
|
| 8489593960 |
d(u v) - v du = u dv
|
|
|
|
|
| 8489593962 |
u dv = d(u v) - v du
|
|
|
|
|
| 8489593964 |
\int u dv = u v - \int v du
|
|
|
|
|
| 8494839423 |
\nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
|
|
|
|
|
| 8572657110 |
1 = \int |\psi(x)|^2 dx
|
|
|
|
|
| 8572852424 |
\vec{E} = E(\vec{r},t)
|
|
|
|
|
| 8575746378 |
\int \frac{1}{2} dx = \frac{1}{2} x
|
|
|
|
|
| 8575748999 |
\frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x)\right) = -k^2 \left(a \sin(kx) + b \cos(kx)\right)
|
|
|
|
|
| 8576785890 |
1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
|
|
|
|
|
| 8577275751 |
0 = a \sin(0) + b\cos(0)
|
|
|
|
|
| 8582885111 |
\psi(x) = a \sin(kx) + b \cos(kx)
|
|
|
|
|
| 8582954722 |
x^2 + 2xh + h^2 = (x+h)^2
|
|
|
|
|
| 8849289982 |
\psi(x)^* = a \sin(\frac{n \pi}{W} x)
|
|
|
|
|
| 8889444440 |
1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right)\right)^2 dx
|
|
|
|
|
| 9059289981 |
\psi(x) = a \sin(k x)
|
|
|
|
|
| 9285928292 |
ax^2 + bx + c = 0
|
|
|
|
|
| 9291999979 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0\vec{\nabla} \times \frac{\partial \vec{H}}{\partial t}
|
|
|
|
|
| 9294858532 |
\hat{A}^+ = \hat{A}
|
|
|
|
|
| 9385938295 |
(x+(b/(2a)))^2 = -(c/a) + (b/(2a))^2
|
|
|
|
|
| 9393939991 |
\psi(x) = -\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
|
|
|
|
|
| 9393939992 |
\psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
|
|
|
|
|
| 9394939493 |
\nabla^2 E(\vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r},t)
|
|
|
|
|
| 9429829482 |
\frac{d}{dx} y = -\sin(x) + i\cos(x)
|
|
|
|
|
| 9482113948 |
\frac{dy}{y} = i dx
|
|
|
|
|
| 9482438243 |
(\cos(x))^2 = \cos(2x)+(\sin(x))^2
|
|
|
|
|
| 9482923849 |
\exp(i x) = y
|
|
|
|
|
| 9482928242 |
\cos(2x) = (\cos(x))^2 - (\sin(x))^2
|
|
|
|
|
| 9482928243 |
\cos(2x)+(\sin(x))^2 = (\cos(x))^2
|
|
|
|
|
| 9482943948 |
\log(y) = i dx
|
|
|
|
|
| 9482948292 |
b = c
|
|
|
|
|
| 9482984922 |
\frac{d}{dx} y = (i\sin(x) + \cos(x)) i
|
|
|
|
|
| 9483928192 |
\cos(2x)+i\sin(2x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
|
|
|
|
|
| 9485384858 |
\nabla^2 E(\vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 9485747245 |
\sqrt{\frac{2}{W}} = a
|
|
|
|
|
| 9485747246 |
-\sqrt{\frac{2}{W}} = a
|
|
|
|
|
| 9492920340 |
y = \cos(x)+i \sin(x)
|
|
|
|
|
| 9494829190 |
k
|
|
|
|
|
| 9495857278 |
\psi(x=W) = 0
|
|
|
|
|
| 9499428242 |
E(\vec{r},t) = E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 9499959299 |
a + k = b + k
|
|
|
|
|
| 9582958293 |
x = \sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
|
|
|
|
|
| 9582958294 |
x+(b/(2a)) = \sqrt{(b/(2a))^2 - (c/a)}
|
|
|
|
|
| 9584821911 |
c + d = a
|
|
|
|
|
| 9585727710 |
\psi(x=0) = 0
|
|
|
|
|
| 9596004948 |
x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle
|
|
|
|
|
| 9848292229 |
dy = y i dx
|
|
|
|
|
| 9848294829 |
\frac{d}{dx} y = y i
|
|
|
|
|
| 9858028950 |
\frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
|
|
|
|
|
| 9889984281 |
2(\sin(x))^2 = 1-\cos(2x)
|
|
|
|
|
| 9919999981 |
\rho = 0
|
|
|
|
|
| 9958485859 |
\frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
|
|
|
|
|
| 9988949211 |
(\sin(x))^2 = \frac{1-\cos(2x)}{2}
|
|
|
|
|
| 9991999979 |
\vec{\nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t}
|
|
|
|
|
| 9999998870 |
\frac{\vec{p}}{\hbar} = \vec{k}
|
|
|
|
|
| 9999999870 |
\frac{p}{\hbar} = k
|
|
|
|
|
| 9999999950 |
\beta = 1/(k_{Boltzmann}T)
|
|
|
|
|
| 9999999951 |
\langle x | k \rangle = \frac{\exp(i k x)}{\sqrt{2\pi}}
|
|
|
|
|
| 9999999952 |
f(x) \delta(x-a) = f(a)
|
|
|
|
|
| 9999999953 |
\int_{-\infty}^{\infty} \delta(x) dx = 1
|
|
|
|
|
| 9999999954 |
c = 1/\sqrt{\epsilon_0 \mu_0}
|
|
|
|
|
| 9999999955 |
\vec{E} = -\vec{\nabla} \Psi
|
|
|
|
|
| 9999999956 |
\vec{F} = \frac{\partial}{\partial t}\vec{p}
|
|
|
|
|
| 9999999957 |
\vec{F} = -\vec{\nabla} V
|
|
|
|
|
| 9999999960 |
\hbar = h/(2 \pi)
|
|
|
|
|
| 9999999961 |
\frac{E}{\hbar} = \omega
|
|
|
|
|
| 9999999962 |
p = \hbar k
|
|
|
|
|
| 9999999963 |
\lambda = h/p
|
|
|
|
|
| 9999999964 |
\omega = c k
|
|
|
|
|
| 9999999965 |
E = \omega \hbar
|
|
|
|
|
| 9999999966 |
\vec{L} = \vec{r}\times\vec{p}
|
|
|
|
|
| 9999999967 |
\vec{S} = \frac{1}{\mu_0} \vec{E}\times \vec{B}
|
|
|
|
|
| 9999999968 |
x = \frac{-b-\sqrt{b^2-4ac}}{2a}
|
|
|
|
|
| 9999999969 |
x = \frac{-b+\sqrt{b^2-4ac}}{2a}
|
|
|
|
|
| 9999999970 |
\eta_1 \sin\theta_1 = \eta_2 \sin\theta_2
|
|
|
|
|
| 9999999971 |
{\cal H} = \frac{p^2}{2m}+V
|
|
|
|
|
| 9999999972 |
{\cal H} |\psi\rangle = E |\psi\rangle
|
|
|
|
|
| 9999999973 |
\left( \Delta A \right)^2 = \langle A^2 \rangle - \langle A \rangle^2
|
|
|
|
|
| 9999999974 |
\langle \psi| \hat{A} |\psi\rangle = \int_{-\infty}^{\infty} \psi^* A \psi dx
|
|
|
|
|
| 9999999975 |
\langle \psi| \hat{A} |\psi\rangle = \langle a \rangle
|
|
|
|
|
| 9999999976 |
\hat{p} = -i \hbar \frac{\partial }{\partial x}
|
|
|
|
|
| 9999999977 |
[\hat{x},\hat{p}] = i \hbar
|
|
|
|
|
| 9999999978 |
\vec{\nabla} \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial E}{\partial t}
|
|
|
|
|
| 9999999979 |
\vec{\nabla} \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}
|
|
|
|
|
| 9999999980 |
\vec{\nabla} \cdot \vec{B} = 0
|
|
|
|
|
| 9999999981 |
\vec{\nabla} \cdot \vec{E} = \rho/\epsilon_0
|
|
|
|
|
| 9999999982 |
V = I R + Q/C + L \frac{\partial I}{\partial t}
|
|
|
|
|
| 9999999983 |
C = V A/d
|
|
|
|
|
| 9999999984 |
Q = C V
|
|
|
|
|
| 9999999985 |
V = I R
|
|
|
|
|
| 9999999986 |
\left[\right]\left[\right] = \left[\right]
|
|
|
|
|
| 9999999987 |
1 atmosphere = 101.325 Pascal
|
|
|
|
|
| 9999999988 |
1 atmosphere = 14.7 pounds/(inch^2)
|
|
|
|
|
| 9999999989 |
mass_{electron} = 511000 electronVolts/(q^2)
|
|
|
|
|
| 9999999990 |
Tesla = 10000*Gauss
|
|
|
|
|
| 9999999991 |
Pascal = Newton / (meter^2)
|
|
|
|
|
| 9999999992 |
Tesla = Newton / (Ampere*meter)
|
|
|
|
|
| 9999999993 |
Farad = Coulumb / Volt
|
|
|
|
|
| 9999999995 |
Volt = Joule / Coulumb
|
|
|
|
|
| 9999999996 |
Coulomb = Ampere / second
|
|
|
|
|
| 9999999997 |
Watt = Joule / second
|
|
|
|
|
| 9999999998 |
Joule = Newton*meter
|
|
|
|
|
| 9999999999 |
Newton = kilogram*meter/(second^2)
|
|
|
|
|
| 4961662865 |
x
|
|
|
|
|
| 9110536742 |
2 x
|
|
|
|
|
| 8483686863 |
\sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x)\right)
|
|
|
|
|
| 3470587782 |
\sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
|
|
|
|
|
| 8642992037 |
2
|
|
|
|
|
| 9894826550 |
2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \left(\exp(i x)+\exp(-i x)\right)
|
|
|
|
|
| 4257214632 |
\frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)
|
|
|
|
|
| 8699789241 |
2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)
|
|
|
|
|
| 9180861128 |
2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)
|
|
|
|
|
| 2405307372 |
\sin(2 x) = 2 \sin(x) \cos(x)
|
|
|
|
|
| 3268645065 |
x
|
|
|
|
|
| 9350663581 |
\pi
|
|
|
|
|
| 8332931442 |
\exp(i \pi) = \cos(\pi)+i \sin(\pi)
|
|
|
|
|
| 6885625907 |
\exp(i \pi) = -1 + i 0
|
|
|
|
|
| 3331824625 |
\exp(i \pi) = -1
|
|
|
|
|
| 4901237716 |
1
|
|
|
|
|
| 2501591100 |
\exp(i \pi) + 1 = 0
|
|
|
|
|
| 5136652623 |
E = KE + PE
|
|
mechanical energy is the sum of the potential plus kinetic energies |
|
|
| 7915321076 |
E, KE, PE
|
|
|
|
|
| 3192374582 |
E_2, KE_2, PE_2
|
|
|
|
|
| 7875206161 |
E_2 = KE_2 + PE_2
|
|
|
|
|
| 4229092186 |
E, KE, PE
|
|
|
|
|
| 1490821948 |
E_1, KE_1, PE_1
|
|
|
|
|
| 4303372136 |
E_1 = KE_1 + PE_1
|
|
|
|
|
| 5514556106 |
E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1)
|
|
|
|
|
| 8357234146 |
KE = \frac{1}{2} m v^2
|
|
kinetic energy |
Kinetic_energy
|
|
| 3658066792 |
KE_2, v_2
|
|
|
|
|
| 8082557839 |
KE, v
|
|
|
|
|
| 7676652285 |
KE_2 = \frac{1}{2} m v_2^2
|
|
|
|
|
| 2790304597 |
KE_1, v_1
|
|
|
|
|
| 9880327189 |
KE, v
|
|
|
|
|
| 4928007622 |
KE_1 = \frac{1}{2} m v_1^2
|
|
|
|
|
| 5733146966 |
KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right)
|
|
|
|
|
| 6554292307 |
t
|
|
|
|
|
| 4270680309 |
\frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t}
|
|
|
|
|
| 5781981178 |
x^2 - y^2 = (x+y)(x-y)
|
|
difference of squares |
Difference_of_two_squares
|
|
| 5946759357 |
v_2, v_1
|
|
|
|
|
| 6675701064 |
x, y
|
|
|
|
|
| 4648451961 |
v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1)
|
|
|
|
|
| 9356924046 |
\frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t}
|
|
|
|
|
| 9397152918 |
v = \frac{v_1 + v_2}{2}
|
|
average velocity |
|
|
| 2857430695 |
a = \frac{v_2 - v_1}{t}
|
|
acceleration |
|
|
| 7735737409 |
\frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t}
|
|
|
|
|
| 4784793837 |
\frac{KE_2 - KE_1}{t} = m v a
|
|
|
|
|
| 5345738321 |
F = m a
|
|
Newton's second law of motion |
Newton%27s_laws_of_motion#Newton's_second_law
|
|
| 2186083170 |
\frac{KE_2 - KE_1}{t} = v F
|
|
|
|
|
| 6715248283 |
PE = -F x
|
|
potential energy |
Potential_energy
|
|
| 3852078149 |
PE_2, x_2
|
|
|
|
|
| 7388921188 |
PE, x
|
|
|
|
|
| 2431507955 |
PE_2 = -F x_2
|
|
|
|
|
| 3412641898 |
PE_1, x_1
|
|
|
|
|
| 9937169749 |
PE, x
|
|
|
|
|
| 4669290568 |
PE_1 = -F x_1
|
|
|
|
|
| 7734996511 |
PE_2 - PE_1 = -F ( x_2 - x_1 )
|
|
|
|
|
| 2016063530 |
t
|
|
|
|
|
| 7267155233 |
\frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right)
|
|
|
|
|
| 9337785146 |
v = \frac{x_2 - x_1}{t}
|
|
average velocity |
|
|
| 4872970974 |
\frac{PE_2 - PE_1}{t} = -F v
|
|
|
|
|
| 2081689540 |
t
|
|
|
|
|
| 2770069250 |
\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t}
|
|
|
|
|
| 3591237106 |
\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v
|
|
|
|
|
| 1772416655 |
\frac{E_2 - E_1}{t} = v F - F v
|
|
|
|
|
| 1809909100 |
\frac{E_2 - E_1}{t} = 0
|
|
|
|
|
| 5778176146 |
t
|
|
|
|
|
| 3806977900 |
E_2 - E_1 = 0
|
|
|
|
|
| 5960438249 |
E_1
|
|
|
|
|
| 8558338742 |
E_2 = E_1
|
|
|
|
|
| 8945218208 |
\theta_{Brewster} + \theta_{refracted} = 90^{\circ}
|
|
|
based on figure 34-27 on page 824 in 2001_HRW
|
|
| 9025853427 |
\theta_{Brewster}
|
|
|
|
|
| 1310571337 |
\theta_{refracted} = 90^{\circ} - \theta_{Brewster}
|
|
|
|
|
| 6450985774 |
n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 )
|
|
Law of Refraction |
eq 34-44 on page 819 in 2001_HRW
|
|
| 1779497048 |
\theta_{Brewster}, \theta_{refraction}
|
|
|
|
|
| 7322344455 |
\theta_1, \theta_2
|
|
|
|
|
| 2575937347 |
n_1 \sin( \theta_{Brewster} ) = n_2 \sin( \theta_{refraction} )
|
|
|
|
|
| 7696214507 |
n_1 \sin( \theta_{Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{Brewster} )
|
|
|
|
|
| 8588429722 |
\sin( 90^{\circ} - x ) = \cos( x )
|
|
|
|
|
| 7375348852 |
\theta_{Brewster}
|
|
|
|
|
| 1512581563 |
x
|
|
|
|
|
| 6831637424 |
\sin( 90^{\circ} - \theta_{Brewster} ) = \cos( \theta_{Brewster} )
|
|
|
|
|
| 3061811650 |
n_1 \sin( \theta_{Brewster} ) = n_2 \cos( \theta_{Brewster} )
|
|
|
|
|
| 7857757625 |
n_1
|
|
|
|
|
| 9756089533 |
\sin( \theta_{Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{Brewster} )
|
|
|
|
|
| 5632428182 |
\cos( \theta_{Brewster} )
|
|
|
|
|
| 2768857871 |
\frac{\sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )} = \frac{n_2}{n_1}
|
|
|
|
|
| 4968680693 |
\tan( x ) = \frac{ \sin( x )}{\cos( x )}
|
|
|
|
|
| 7321695558 |
\theta_{Brewster}
|
|
|
|
|
| 9906920183 |
x
|
|
|
|
|
| 4501377629 |
\tan( \theta_{Brewster} ) = \frac{ \sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )}
|
|
|
|
|
| 3417126140 |
\tan( \theta_{Brewster} ) = \frac{ n_2 }{ n_1 }
|
|
|
|
|
| 5453995431 |
\arctan{ x }
|
|
|
|
|
| 6023986360 |
x
|
|
|
|
|
| 8495187962 |
\theta_{Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }
|
|
|
|
|
| 9040079362 |
f
|
|
|
|
|
| 9565166889 |
T
|
|
|
|
|
| 5426308937 |
v = \frac{d}{t}
|
|
|
|
|
| 7476820482 |
C
|
|
|
|
|
| 1277713901 |
d
|
|
|
|
|
| 6946088325 |
v = \frac{C}{t}
|
|
|
|
|
| 6785303857 |
C = 2 \pi r
|
|
|
|
|
| 4816195267 |
C, r
|
|
|
|
|
| 2113447367 |
C_{\rm{Earth\ orbit}}, r_{\rm{Earth\ orbit}}
|
|
|
|
|
| 6348260313 |
C_{\rm{Earth\ orbit}} = 2 \pi r_{\rm{Earth\ orbit}}
|
|
|
|
|
| 3847777611 |
C, v, t
|
|
|
|
|
| 6406487188 |
C_{\rm{Earth\ orbit}}, v_{\rm{Earth\ orbit}}, t_{\rm{Earth\ orbit}}
|
|
|
|
|
| 3046191961 |
v_{\rm{Earth\ orbit}} = \frac{C_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
|
|
|
|
|
| 3080027960 |
v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
|
|
|
|
|
| 7175416299 |
t_{\rm{Earth\ orbit}} = 1 {\rm{year}}
|
|
|
|
|
| 3219318145 |
\frac{365 {\rm days}}{1 {\rm year}} \frac{24 {\rm hours}}{1 {\rm day}} \frac{60 {\rm minutes}}{1 {\rm hour}} \frac{60 {\rm seconds}}{1 {\rm minute}}
|
|
|
|
|
| 8721295221 |
t_{\rm{Earth\ orbit}} = 3.16 10^7 {\rm{seconds}}
|
|
|
|
|
| 4593428198 |
v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{3.16\ 10^7 {\rm seconds}}
|
|
|
|
|
| 3472836147 |
r_{\rm{Earth\ orbit}} = 1.496\ 10^8 {\rm km}
|
|
|
|
|
| 6998364753 |
v_{\rm{Earth\ orbit}} = \frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}}
|
|
|
|
|
| 4180845508 |
v_{\rm{Earth\ orbit}} = 29.8 \frac{{\rm km}}{{\rm sec}}
|
|
|
|
|
| 4662369843 |
x' = \gamma (x - v t)
|
|
|
|
|
| 2983053062 |
x = \gamma (x' + v t')
|
|
|
|
|
| 3426941928 |
x = \gamma ( \gamma (x - v t) + v t' )
|
|
|
|
|
| 2096918413 |
x = \gamma ( \gamma x - \gamma v t + v t' )
|
|
|
|
|
| 7741202861 |
x = \gamma^2 x - \gamma^2 v t + \gamma v t'
|
|
|
|
|
| 7337056406 |
\gamma^2 x
|
|
|
|
|
| 4139999399 |
x - \gamma^2 x = - \gamma^2 v t + \gamma v t'
|
|
|
|
|
| 3495403335 |
x
|
|
|
|
|
| 9031609275 |
x (1 - \gamma^2 ) = - \gamma^2 v t + \gamma v t'
|
|
|
|
|
| 8014566709 |
\gamma^2 v t
|
|
|
|
|
| 9409776983 |
x (1 - \gamma^2 ) + \gamma^2 v t = \gamma v t'
|
|
|
|
|
| 2226340358 |
\gamma v
|
|
|
|
|
| 6417705759 |
\frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = \gamma v t'
|
|
|
|
|
| 8730201316 |
\frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t = t'
|
|
|
first term was multiplied by \gamma/\gamma
|
|
| 1974334644 |
\frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = t'
|
|
|
|
|
| 5148266645 |
t' = \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t
|
|
|
|
|
| 4287102261 |
x^2 + y^2 + z^2 = c^2 t^2
|
|
|
describes a spherical wavefront
|
|
| 1201689765 |
x'^2 + y'^2 + z'^2 = c^2 t'^2
|
|
|
describes a spherical wavefront for an observer in a moving frame of reference
|
|
| 7057864873 |
y' = y
|
|
|
frame of reference is moving only along x direction
|
|
| 8515803375 |
z' = z
|
|
|
frame of reference is moving only along x direction
|
|
| 6153463771 |
3
|
|
|
|
|
| 4746576736 |
asdf
|
|
|
|
|
| 1027270623 |
minga
|
|
|
|
|
| 6101803533 |
ming
|
|
|
|
|
| 9231495607 |
a = b
|
|
|
|
|
| 8664264194 |
b + 2
|
|
|
|
|
| 9805063945 |
\gamma^2 (x - v t)^2 + y^2 + z^2 = c^2 \gamma^2 \left( t + \frac{ 1 - \gamma^2 }{ \gamma^2 } \frac{x}{v} \right)^2
|
|
|
|
|
| 4057686137 |
C \to C_{\rm{Earth\ orbit}}
|
|
|
|
|
| 2346150725 |
r \to r_{\rm{Earth\ orbit}}
|
|
|
|
|
| 9753878784 |
v \to v_{\rm{Earth\ orbit}}
|
|
|
|
|
| 8135396036 |
t \to t_{\rm{Earth\ orbit}}
|
|
|
|
|
| 2773628333 |
\theta_1 \to \theta_{Brewster}
|
|
|
|
|
| 7154592211 |
\theta_2 \to \theta_{\rm refracted}
|
|
|
|
|
| 3749492596 |
E \to E_1
|
|
|
|
|
| 1258245373 |
E \to E_2
|
|
|
|
|
| 4147101187 |
KE \to KE_1
|
|
|
|
|
| 3809726424 |
PE \to PE_1
|
|
|
|
|
| 6383056612 |
KE \to KE_2
|
|
|
|
|
| 5075406409 |
PE \to PE_2
|
|
|
|
|
| 5398681503 |
v \to v_1
|
|
|
|
|
| 9305761407 |
v \to v_2
|
|
|
|
|
| 4218009993 |
x \to x_1
|
|
|
|
|
| 4188639044 |
x \to x_2
|
|
|
|
|
| 8173074178 |
x \to v_2
|
|
|
|
|
| 1025759423 |
y \to v_1
|
|
|
|
|
| 1935543849 |
\gamma^2 x^2 - \gamma^2 2 x v t + \gamma^2 v^2 t^2 + y^2 + z^2 = c^2 \gamma^2 \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x^2}{\gamma^2} + c^2 \gamma^2 2 t \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x}{\gamma} + c^2 \gamma^2 t^2
|
|
|
|
|
| 1586866563 |
\left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right) = t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)
|
|
|
|
|
| 3182633789 |
\gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1
|
|
|
|
|
| 1916173354 |
-\gamma^2 v^2 + c^2 \gamma^2 = c^2
|
|
|
|
|
| 2076171250 |
-\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} = 0
|
|
|
|
|
| 5284610349 |
\gamma^2
|
|
|
|
|
| 2417941373 |
- c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 - \gamma^2
|
|
|
|
|
| 5787469164 |
1 - \gamma^2
|
|
|
|
|
| 1639827492 |
- c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2} = 1
|
|
|
|
|
| 5669500954 |
v^2 \gamma^2
|
|
|
|
|
| 5763749235 |
-c^2 + c^2 \gamma^2 = v^2 \gamma^2
|
|
|
|
|
| 6408214498 |
c^2
|
|
|
|
|
| 2999795755 |
c^2 \gamma^2 = v^2 \gamma^2 + c^2
|
|
|
|
|
| 3412946408 |
v^2 \gamma^2
|
|
|
|
|
| 2542420160 |
c^2 \gamma^2 - v^2 \gamma^2 = c^2
|
|
|
|
|
| 7743841045 |
\gamma^2
|
|
|
|
|
| 7513513483 |
\gamma^2 (c^2 - v^2) = c^2
|
|
|
|
|
| 8571466509 |
c^2 - \gamma^2
|
|
|
|
|
| 7906112355 |
\gamma^2 = \frac{c^2}{c^2 - \gamma^2}
|
|
|
|
|
| 3060139639 |
1/2
|
|
|
|
|
| 1528310784 |
\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
|
|
|
|
|
| 8360117126 |
\gamma = \frac{-1}{\sqrt{1-\frac{v^2}{c^2}}}
|
|
|
not a physically valid result in this context
|
|
| 3366703541 |
a = \frac{v - v_0}{t}
|
|
|
acceleration is the average change in speed over a duration
|
|
| 7083390553 |
t
|
|
|
|
|
| 4748157455 |
a t = v - v_0
|
|
|
|
|
| 6417359412 |
v_0
|
|
|
|
|
| 4798787814 |
a t + v_0 = v
|
|
|
|
|
| 3462972452 |
v = v_0 + a t
|
|
|
|
|
| 3411994811 |
v_{\rm ave} = \frac{d}{t}
|
|
|
|
|
| 6175547907 |
v_{\rm ave} = \frac{v + v_0}{2}
|
|
|
|
|
| 9897284307 |
\frac{d}{t} = \frac{v + v_0}{2}
|
|
|
|
|
| 8865085668 |
t
|
|
|
|
|
| 8706092970 |
d = \left(\frac{v + v_0}{2}\right)t
|
|
|
|
|
| 7011114072 |
d = \frac{(v_0 + a t) + v_0}{2} t
|
|
|
|
|
| 1265150401 |
d = \frac{2 v_0 + a t}{2} t
|
|
|
|
|
| 9658195023 |
d = v_0 t + \frac{1}{2} a t^2
|
|
|
|
|
| 5799753649 |
2
|
|
|
|
|
| 7215099603 |
v^2 = v_0^2 + 2 a t v_0 + a^2 t^2
|
|
|
|
|
| 5144263777 |
v^2 = v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right)
|
|
|
|
|
| 7939765107 |
v^2 = v_0^2 + 2 a d
|
|
|
|
|
| 6729698807 |
v_0
|
|
|
|
|
| 9759901995 |
v - v_0 = a t
|
|
|
|
|
| 2242144313 |
a
|
|
|
|
|
| 1967582749 |
t = \frac{v - v_0}{a}
|
|
|
|
|
| 5733721198 |
d = \frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right)
|
|
|
|
|
| 5611024898 |
d = \frac{1}{2 a} (v^2 - v_0^2)
|
|
|
|
|
| 5542390646 |
2 a
|
|
|
|
|
| 8269198922 |
2 a d = v^2 - v_0^2
|
|
|
|
|
| 9070454719 |
v_0^2
|
|
|
|
|
| 4948763856 |
2 a d + v_0^2 = v^2
|
|
|
|
|
| 9645178657 |
a t
|
|
|
|
|
| 6457044853 |
v - a t = v_0
|
|
|
|
|
| 1259826355 |
d = (v - a t) t + \frac{1}{2} a t^2
|
|
|
|
|
| 4580545876 |
d = v t - a t^2 + \frac{1}{2} a t^2
|
|
|
|
|
| 6421241247 |
d = v t - \frac{1}{2} a t^2
|
|
|
|
|
| 4182362050 |
Z = |Z| \exp( i \theta )
|
|
|
Z \in \mathbb{C}
|
|
| 1928085940 |
Z^* = |Z| \exp( -i \theta )
|
|
|
|
|
| 9191880568 |
Z Z^* = |Z| |Z| \exp( -i \theta ) \exp( i \theta )
|
|
|
|
|
| 3350830826 |
Z Z^* = |Z|^2
|
|
|
|
|
| 8396997949 |
I = | A + B |^2
|
|
|
intensity of two waves traveling opposite directions on same path
|
|
| 4437214608 |
Z
|
|
|
|
|
| 5623794884 |
A + B
|
|
|
|
|
| 2236639474 |
(A + B)(A + B)^* = |A + B|^2
|
|
|
|
|
| 1020854560 |
I = (A + B)(A + B)^*
|
|
|
|
|
| 6306552185 |
I = (A + B)(A^* + B^*)
|
|
|
|
|
| 8065128065 |
I = A A^* + B B^* + A B^* + B A^*
|
|
|
|
|
| 9761485403 |
Z
|
|
|
|
|
| 8710504862 |
A
|
|
|
|
|
| 4075539836 |
A A^* = |A|^2
|
|
|
|
|
| 6529120965 |
B
|
|
|
|
|
| 1511199318 |
Z
|
|
|
|
|
| 7107090465 |
B B^* = |B|^2
|
|
|
|
|
| 5125940051 |
I = |A|^2 + B B^* + A B^* + B A^*
|
|
|
|
|
| 1525861537 |
I = |A|^2 + |B|^2 + A B^* + B A^*
|
|
|
|
|
| 1742775076 |
Z \to B
|
|
|
|
|
| 7607271250 |
\theta \to \phi
|
|
|
|
|
| 4192519596 |
B = |B| \exp(i \phi)
|
|
|
|
|
| 2064205392 |
A
|
|
|
|
|
| 1894894315 |
Z
|
|
|
|
|
| 1357848476 |
A = |A| \exp(i \theta)
|
|
|
|
|
| 6555185548 |
A^* = |A| \exp(-i \theta)
|
|
|
|
|
| 4504256452 |
B^* = |B| \exp(-i \phi)
|
|
|
|
|
| 7621705408 |
I = |A|^2 + |B|^2 + |A| |B| \exp(-i \theta) \exp(i \phi) + |A| |B| \exp(i \theta) \exp(-i \phi)
|
|
|
|
|
| 3085575328 |
I = |A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi))
|
|
|
|
|
| 4935235303 |
x
|
|
|
|
|
| 2293352649 |
\theta - \phi
|
|
|
|
|
| 3660957533 |
\cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
|
|
|
|
|
| 3967985562 |
2
|
|
|
|
|
| 2700934933 |
2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
|
|
|
|
|
| 8497631728 |
I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )
|
|
|
|
|
| 6774684564 |
\theta = \phi
|
|
|
for coherent waves
|
|
| 2099546947 |
I = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )
|
|
|
|
|
| 2719691582 |
|A| = |B|
|
|
|
in a loop
|
|
| 3257276368 |
I = |A|^2 + |A|^2 + |A| |A| 2
|
|
|
|
|
| 8283354808 |
I_{\rm coherent} = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )
|
|
|
|
|
| 8046208134 |
I_{\rm coherent} = |A|^2 + |A|^2 + |A| |A| 2
|
|
|
|
|
| 1172039918 |
I_{\rm coherent} = 4 |A|^2
|
|
|
|
|
| 8602221482 |
\langle \cos(\theta - \phi) \rangle = 0
|
|
|
incoherent light source
|
|
| 6240206408 |
I_{\rm incoherent} = |A|^2 + |B|^2
|
|
|
|
|
| 6529793063 |
I_{\rm incoherent} = |A|^2 + |A|^2
|
|
|
|
|
| 3060393541 |
I_{\rm incoherent} = 2|A|^2
|
|
|
|
|
| 6556875579 |
\frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2
|
|
|
|
|
| 8750500372 |
f
|
|
|
|
|
| 7543102525 |
g
|
|
|
|
|
| 8267133829 |
a = b
|
|
|
|
|
| 7968822469 |
c = d
|
|
|
|
|
| 3169580383 |
\vec{a} = \frac{d\vec{v}}{dt}
|
|
|
acceleration is the change in speed over a duration
|
|
| 8602512487 |
\vec{a} = a_x \hat{x} + a_y \hat{y}
|
|
|
decompose acceleration into two components
|
|
| 4158986868 |
a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt}
|
|
|
|
|
| 5349866551 |
\vec{v} = v_x \hat{x} + v_y \hat{y}
|
|
|
|
|
| 7729413831 |
a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right)
|
|
|
|
|
| 1819663717 |
a_x = \frac{d}{dt} v_x
|
|
|
|
|
| 8228733125 |
a_y = \frac{d}{dt} v_y
|
|
|
|
|
| 9707028061 |
a_x = 0
|
|
|
|
|
| 2741489181 |
a_y = -g
|
|
|
|
|
| 3270039798 |
2
|
|
|
|
|
| 8880467139 |
2
|
|
|
|
|
| 8750379055 |
0 = \frac{d}{dt} v_x
|
|
|
|
|
| 1977955751 |
-g = \frac{d}{dt} v_y
|
|
|
|
|
| 6672141531 |
dt
|
|
|
|
|
| 1702349646 |
-g dt = d v_y
|
|
|
|
|
| 8584698994 |
-g \int dt = \int d v_y
|
|
|
|
|
| 9973952056 |
-g t = v_y - v_{0, y}
|
|
|
|
|
| 4167526462 |
v_{0, y}
|
|
|
|
|
| 6572039835 |
-g t + v_{0, y} = v_y
|
|
|
|
|
| 7252338326 |
v_y = \frac{dy}{dt}
|
|
|
|
|
| 6204539227 |
-g t + v_{0, y} = \frac{dy}{dt}
|
|
|
|
|
| 1614343171 |
dt
|
|
|
|
|
| 8145337879 |
-g t dt + v_{0, y} dt = dy
|
|
|
|
|
| 8808860551 |
-g \int t dt + v_{0, y} \int dt = \int dy
|
|
|
|
|
| 2858549874 |
- \frac{1}{2} g t^2 + v_{0, y} t = y - y_0
|
|
|
|
|
| 6098638221 |
y_0
|
|
|
|
|
| 2461349007 |
- \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y
|
|
|
|
|
| 8717193282 |
dt
|
|
|
|
|
| 1166310428 |
0 dt = d v_x
|
|
|
|
|
| 2366691988 |
\int 0 dt = \int d v_x
|
|
|
|
|
| 1676472948 |
0 = v_x - v_{0, x}
|
|
|
|
|
| 1439089569 |
v_{0, x}
|
|
|
|
|
| 6134836751 |
v_{0, x} = v_x
|
|
|
|
|
| 8460820419 |
v_x = \frac{dx}{dt}
|
|
|
|
|
| 7455581657 |
v_{0, x} = \frac{dx}{dt}
|
|
|
|
|
| 8607458157 |
dt
|
|
|
|
|
| 1963253044 |
v_{0, x} dt = dx
|
|
|
|
|
| 3676159007 |
v_{0, x} \int dt = \int dx
|
|
|
|
|
| 9882526611 |
v_{0, x} t = x - x_0
|
|
|
|
|
| 3182907803 |
x_0
|
|
|
|
|
| 8486706976 |
v_{0, x} t + x_0 = x
|
|
|
|
|
| 1306360899 |
x = v_{0, x} t + x_0
|
|
|
|
|
| 7049769409 |
2
|
|
|
|
|
| 9341391925 |
\vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y}
|
|
|
|
|
| 6410818363 |
\theta
|
|
|
|
|
| 7391837535 |
\cos(\theta) = \frac{v_{0, x}}{v_0}
|
|
|
|
|
| 7376526845 |
\sin(\theta) = \frac{v_{0, y}}{v_0}
|
|
|
|
|
| 5868731041 |
v_0
|
|
|
|
|
| 6083821265 |
v_0 \cos(\theta) = v_{0, x}
|
|
|
|
|
| 5438722682 |
x = v_0 t \cos(\theta) + x_0
|
|
|
|
|
| 5620558729 |
v_0
|
|
|
|
|
| 8949329361 |
v_0 \sin(\theta) = v_{0, y}
|
|
|
|
|
| 1405465835 |
y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0
|
|
|
|
|
| 9862900242 |
y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0
|
|
|
|
|
| 6050070428 |
v_{0, x}
|
|
|
|
|
| 3274926090 |
t = \frac{x - x_0}{v_{0, x}}
|
|
|
|
|
| 7354529102 |
y = - \frac{1}{2} g \left( \frac{x - x_0}{v_{0, x}} \right)^2 + v_{0, y} \frac{x - x_0}{v_{0, x}} + y_0
|
|
|
|
|
| 8406170337 |
y \to y_f
|
|
|
|
|
| 2403773761 |
t \to t_f
|
|
|
|
|
| 5379546684 |
y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
|
|
|
|
|
| 5373931751 |
t = t_f
|
|
|
|
|
| 9112191201 |
y_f = 0
|
|
|
|
|
| 8198310977 |
0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
|
|
|
|
|
| 1650441634 |
y_0 = 0
|
|
|
define coordinate system such that initial height is at origin
|
|
| 1087417579 |
0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)
|
|
|
|
|
| 4829590294 |
t_f
|
|
|
|
|
| 2086924031 |
0 = - \frac{1}{2} g t_f + v_0 \sin(\theta)
|
|
|
|
|
| 6974054946 |
\frac{1}{2} g t_f
|
|
|
|
|
| 1191796961 |
\frac{1}{2} g t_f = v_0 \sin(\theta)
|
|
|
|
|
| 2510804451 |
2/g
|
|
|
|
|
| 4778077984 |
t_f = \frac{2 v_0 \sin(\theta)}{g}
|
|
|
|
|
| 3273630811 |
x \to x_f
|
|
|
|
|
| 6732786762 |
t \to t_f
|
|
|
|
|
| 3485125659 |
x_f = v_0 t_f \cos(\theta) + x_0
|
|
|
|
|
| 4370074654 |
t = t_f
|
|
|
|
|
| 2378095808 |
x_f = x_0 + d
|
|
|
|
|
| 4268085801 |
x_0 + d = v_0 t_f \cos(\theta) + x_0
|
|
|
|
|
| 8072682558 |
x_0
|
|
|
|
|
| 7233558441 |
d = v_0 t_f \cos(\theta)
|
|
|
|
|
| 2297105551 |
d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)
|
|
|
|
|
| 7587034465 |
\theta
|
|
|
|
|
| 7214442790 |
x
|
|
|
|
|
| 2519058903 |
\sin(2 \theta) = 2 \sin(\theta) \cos(\theta)
|
|
|
|
|
| 8922441655 |
d = \frac{v_0^2}{g} \sin(2 \theta)
|
|
|
|
|
| 5667870149 |
\theta
|
|
|
|
|
| 1541916015 |
\theta = \frac{\pi}{4}
|
|
|
|
|
| 3607070319 |
d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)
|
|
|
|
|
| 5353282496 |
d = \frac{v_0^2}{g}
|
|
|
|
|
| 0000040490 |
a^2
|
|
|
|
|
| 0000999900 |
b/(2a)
|
|
|
|
|
| 0001030901 |
\cos(x)
|
|
|
|
|
| 0001111111 |
(\sin(x))^2
|
|
|
|
|
| 0001209482 |
2 \pi
|
|
|
|
|
| 0001304952 |
\hbar
|
|
|
|
|
| 0001334112 |
W
|
|
|
|
|
| 0001921933 |
2 i
|
|
|
|
|
| 0002239424 |
2
|
|
|
|
|
| 0002338514 |
\vec{p}_{2}
|
|
|
|
|
| 0002342425 |
m/m
|
|
|
|
|
| 0002367209 |
1/2
|
|
|
|
|
| 0002393922 |
x
|
|
|
|
|
| 0002424922 |
a
|
|
|
|
|
| 0002436656 |
i \hbar
|
|
|
|
|
| 0002449291 |
b/(2a)
|
|
|
|
|
| 0002838490 |
b/(2a)
|
|
|
|
|
| 0002919191 |
\sin(-x)
|
|
|
|
|
| 0002929944 |
1/2
|
|
|
|
|
| 0002940021 |
2 \pi
|
|
|
|
|
| 0003232242 |
t
|
|
|
|
|
| 0003413423 |
\cos(-x)
|
|
|
|
|
| 0003747849 |
-1
|
|
|
|
|
| 0003838111 |
2
|
|
|
|
|
| 0003919391 |
x
|
|
|
|
|
| 0003949052 |
-x
|
|
|
|
|
| 0003949921 |
\hbar
|
|
|
|
|
| 0003954314 |
dx
|
|
|
|
|
| 0003981813 |
-\sin(x)
|
|
|
|
|
| 0004089571 |
2x
|
|
|
|
|
| 0004264724 |
y
|
|
|
|
|
| 0004307451 |
(b/(2a))^2
|
|
|
|
|
| 0004582412 |
x
|
|
|
|
|
| 0004829194 |
2
|
|
|
|
|
| 0004831494 |
a
|
|
|
|
|
| 0004849392 |
x
|
|
|
|
|
| 0004858592 |
h
|
|
|
|
|
| 0004934845 |
x
|
|
|
|
|
| 0004948585 |
a
|
|
|
|
|
| 0005395034 |
a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
|
|
|
|
|
| 0005626421 |
t
|
|
|
|
|
| 0005749291 |
f
|
|
|
|
|
| 0005938585 |
\frac{-\hbar^2}{2m}
|
|
|
|
|
| 0006466214 |
(\sin(x))^2
|
|
|
|
|
| 0006544644 |
t
|
|
|
|
|
| 0006563727 |
x
|
|
|
|
|
| 0006644853 |
c/a
|
|
|
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| 0006656532 |
e
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| 0007471778 |
2(\sin(x))^2
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| 0007563791 |
i
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| 0007636749 |
x
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| 0007894942 |
(\sin(x))^2
|
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| 0008837284 |
T
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| 0008842811 |
\cos(2x)
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| 0009458842 |
\psi(x)
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| 0009484724 |
\frac{n \pi}{W}x
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| 0009485857 |
a^2\frac{2}{W}
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| 0009485858 |
\frac{2n\pi}{W}
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| 0009492929 |
v du
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| 0009587738 |
\psi
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| 0009594995 |
1/2
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| 0009877781 |
y
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| 0203024440 |
1 = \int_0^W a \sin(\frac{n \pi}{W} x) \psi(x)^* dx
|
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|
|
| 0404050504 |
\lambda = \frac{v}{f}
|
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| 0439492440 |
\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)\right|_0^W
|
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| 0934990943 |
k = \frac{2 \pi}{v T}
|
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| 0948572140 |
\int \cos(a x) dx = \frac{1}{a}\sin(a x)
|
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| 1010393913 |
\langle \psi| \hat{A}^+ |\psi \rangle = \langle a \rangle^*
|
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| 1010393944 |
x = \langle\psi_{\alpha}| a_{\beta} |\psi_{\beta} \rangle
|
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| 1010923823 |
k W = n \pi
|
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| 1020010291 |
0 = a \sin(k W)
|
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| 1020394900 |
p = h/\lambda
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| 1020394902 |
E = h f
|
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| 1029039903 |
p = m v
|
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| 1029039904 |
p^2 = m^2 v^2
|
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| 1158485859 |
\frac{-\hbar^2}{2m} \nabla^2 = {\cal H}
|
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| 1202310110 |
\frac{1}{a^2} = \int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
|
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| 1202312210 |
\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
|
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| 1203938249 |
a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle = a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
|
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|
| 1248277773 |
\cos(2x) = 1-2(\sin(x))^2
|
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| 1293913110 |
0 = b
|
|
|
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| 1293923844 |
\lambda = v T
|
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|
| 1314464131 |
\vec{\nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
|
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| 1314864131 |
\vec{\nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E}
|
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| 1395858355 |
x = \langle \psi_{\alpha}| a_{\alpha} |\psi_{\beta}\rangle
|
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| 1492811142 |
f = x - d
|
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| 1492842000 |
\nabla \vec{x} = f(y)
|
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| 1636453295 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = - \nabla^2 \vec{E}
|
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|
|
| 1638282134 |
\vec{p}_{before} = \vec{p}_{after}
|
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|
|
|
| 1648958381 |
\nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{\nabla}\psi(\vec{r},t)\right)
|
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|
|
| 1857710291 |
0 = a \sin(n \pi)
|
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|
| 1858578388 |
\nabla^2 E(\vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 1858772113 |
k = \frac{n \pi}{W}
|
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|
| 1934748140 |
\int |\psi(x)|^2 dx = 1
|
|
|
|
|
| 2029293929 |
\nabla^2 E(\vec{r})\exp(i \omega t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r})\exp(i \omega t)
|
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|
|
| 2103023049 |
\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right)
|
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| 2113211456 |
f = 1/T
|
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|
|
|
| 2123139121 |
-\exp(-i x) = -\cos(x)+i \sin(x)
|
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|
|
|
| 2131616531 |
T f = 1
|
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|
|
|
| 2258485859 |
{\cal H} \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
|
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|
|
| 2394240499 |
x = a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle
|
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| 2394853829 |
\exp(-i x) = \cos(-x)+i \sin(-x)
|
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|
| 2394935831 |
( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
|
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|
| 2394935835 |
\left(\langle\psi| \hat{A} |\psi\rangle \right)^+ = \left(\langle a \rangle\right)^+
|
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|
|
|
| 2395958385 |
\nabla^2 \psi\left(\vec{r},t)\right) = \frac{-p^2}{\hbar} \psi(\vec{r},t)
|
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|
|
|
| 2404934990 |
\langle x^2\rangle -2\langle x \rangle\langle x \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
|
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|
|
|
| 2494533900 |
\langle x^2\rangle -\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
|
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|
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|
| 2569154141 |
\vec{\nabla} \times \frac{\partial}{\partial t}\vec{H} = \epsilon_0 \frac{\partial^2 }{\partial t^2}\vec{E}
|
|
|
|
|
| 2648958382 |
\nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p}\psi(\vec{r},t)\right)
|
|
|
|
|
| 2848934890 |
\langle a \rangle^* = \langle a \rangle
|
|
|
|
|
| 2944838499 |
\psi(x) = a \sin(\frac{n \pi}{W} x)
|
|
|
|
|
| 3121234211 |
\frac{k}{2\pi} = \lambda
|
|
|
|
|
| 3121234212 |
p = \frac{h k}{2\pi}
|
|
|
|
|
| 3121513111 |
k = \frac{2 \pi}{\lambda}
|
|
|
|
|
| 3131111133 |
T = 1 / f
|
|
|
|
|
| 3131211131 |
\omega = 2 \pi f
|
|
|
|
|
| 3132131132 |
\omega = \frac{2\pi}{T}
|
|
|
|
|
| 3147472131 |
\frac{\omega}{2 \pi} = f
|
|
|
|
|
| 3285732911 |
(\cos(x))^2 = 1-(\sin(x))^2
|
|
|
|
|
| 3485475729 |
\nabla^2 E(\vec{r}) = - \frac{\omega^2}{c^2} E(\vec{r})
|
|
|
|
|
| 3585845894 |
\langle \left(x-\langle x \rangle\right)^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 3829492824 |
\frac{1}{2}\left(\exp(i x)+\exp(-i x)\right) = \cos(x)
|
|
|
|
|
| 3924948349 |
a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
|
|
|
|
|
| 3942849294 |
\exp(i x)-\exp(-i x) = 2 i \sin(x)
|
|
|
|
|
| 3943939590 |
x = a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle
|
|
|
|
|
| 3947269979 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
|
|
|
|
|
| 3948571256 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi(\vec{r},t)
|
|
|
|
|
| 3948572230 |
\vec{\nabla}\psi(\vec{r},t) = \psi_0 \vec{\nabla}\exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
|
|
|
|
|
| 3948574224 |
\psi(\vec{r},t) = \psi_0 \exp\left(i\left(\vec{k}\cdot\vec{r} - \omega t \right) \right)
|
|
|
|
|
| 3948574226 |
\psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right)
|
|
|
|
|
| 3948574228 |
\psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
|
|
|
|
|
| 3948574230 |
\psi(\vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
|
|
|
|
|
| 3948574233 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
|
|
|
|
|
| 3948574235 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
|
|
|
|
|
| 3951205425 |
\vec{p}_{after} = \vec{p}_{1}
|
|
|
|
|
| 4147472132 |
E = \frac{h \omega}{2 \pi}
|
|
|
|
|
| 4298359835 |
E = \frac{1}{2}m v^2
|
|
|
|
|
| 4298359845 |
E = \frac{1}{2m}m^2 v^2
|
|
|
|
|
| 4298359851 |
E = \frac{p^2}{2m}
|
|
|
|
|
| 4341171256 |
i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{p^2}{2 m} \psi(\vec{r},t)
|
|
|
|
|
| 4348571256 |
\frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi(\vec{r},t)
|
|
|
|
|
| 4394958389 |
\vec{\nabla}\cdot \left(\vec{\nabla}\psi(\vec{r},t)\right) = \frac{i}{\hbar} \vec{\nabla}\cdot\left( \vec{p} \psi(\vec{r},t)\right)
|
|
|
|
|
| 4585828572 |
\epsilon_0 \mu_0 = \frac{1}{c^2}
|
|
|
|
|
| 4585932229 |
\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
|
|
|
|
|
| 4598294821 |
\exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
|
|
|
|
|
| 4638429483 |
\exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))
|
|
|
|
|
| 4742644828 |
\exp(i x)+\exp(-i x) = 2 \cos(x)
|
|
|
|
|
| 4827492911 |
\cos(2x)+(\sin(x))^2 = 1-(\sin(x))^2
|
|
|
|
|
| 4838429483 |
\exp(2 i x) = \cos(2 x)+i \sin(2 x)
|
|
|
|
|
| 4843995999 |
\frac{1}{2 i}\left(\exp(i x)-\exp(-i x)\right) = \sin(x)
|
|
|
|
|
| 4857472413 |
1 = \int \psi(x)\psi(x)^* dx
|
|
|
|
|
| 4857475848 |
\frac{1}{a^2} = \frac{W}{2}
|
|
|
|
|
| 4858429483 |
\exp(i x)\exp(i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))
|
|
|
|
|
| 4923339482 |
i x = \log(y)
|
|
|
|
|
| 4928239482 |
\log(y) = i x
|
|
|
|
|
| 4928923942 |
a = b
|
|
|
|
|
| 4929218492 |
a+b = c
|
|
|
|
|
| 4929482992 |
b = 2
|
|
|
|
|
| 4938429482 |
\exp(-i x) = \cos(x)+i \sin(-x)
|
|
|
|
|
| 4938429483 |
\exp(i x) = \cos(x)+i \sin(x)
|
|
|
|
|
| 4938429484 |
\exp(-i x) = \cos(x)-i \sin(x)
|
|
|
|
|
| 4943571230 |
\vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
|
|
|
|
|
| 4948934890 |
\langle \psi| \hat{A} |\psi\rangle = \langle a \rangle^*
|
|
|
|
|
| 4949359835 |
\langle x^2\rangle -2\langle x^2 \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 4954839242 |
\cos(2x)+i\sin(2x) = (\cos(x)+i \sin(x))(\cos(x)+i \sin(x))
|
|
|
|
|
| 4978429483 |
\exp(-i x) = \cos(-x)+i \sin(-x)
|
|
|
|
|
| 4984892984 |
a+2 = c
|
|
|
|
|
| 4985825552 |
\nabla^2 E(\vec{r})\exp(i \omega t) = i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 5530148480 |
\vec{p}_{1}-\vec{p}_{2} = \vec{p}_{electron}
|
|
|
|
|
| 5727578862 |
\frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)
|
|
|
|
|
| 5832984291 |
(\sin(x))^2 + (\cos(x))^2 = 1
|
|
|
|
|
| 5857434758 |
\int a dx = a x
|
|
|
|
|
| 5868688585 |
\frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = \frac{p^2}{2m} \psi(\vec{r},t)
|
|
|
|
|
| 5900595848 |
k = \frac{\omega}{v}
|
|
|
|
|
| 5928285821 |
x^2 + 2x(b/(2a)) + (b/(2a))^2 = (x+(b/(2a)))^2
|
|
|
|
|
| 5928292841 |
x^2 + (b/a)x + (b/(2a))^2 = -c/a + (b/(2a))^2
|
|
|
|
|
| 5938459282 |
x^2 + (b/a)x = -c/a
|
|
|
|
|
| 5958392859 |
x^2 + (b/a)x+(c/a) = 0
|
|
|
|
|
| 5959282914 |
x^2 + x(b/a) + (b/(2a))^2 = (x+(b/(2a)))^2
|
|
|
|
|
| 5982958248 |
x = -\sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
|
|
|
|
|
| 5982958249 |
x+(b/(2a)) = -\sqrt{(b/(2a))^2 - (c/a)}
|
|
|
|
|
| 5985371230 |
\vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi(\vec{r},t)
|
|
|
|
|
| 6742123016 |
\vec{p}_{electron}\cdot\vec{p}_{electron} = (\vec{p}_{1}\cdot\vec{p}_{1})+(\vec{p}_{2}\cdot\vec{p}_{2})-2(\vec{p}_{1}\cdot\vec{p}_{2})
|
|
|
|
|
| 7466829492 |
\vec{\nabla} \cdot \vec{E} = 0
|
|
|
|
|
| 7564894985 |
\int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)
|
|
|
|
|
| 7572664728 |
\cos(2x)+2(\sin(x))^2 = 1
|
|
|
|
|
| 7575738420 |
\left(\sin\left(\frac{n \pi}{W}x\right)\right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2}
|
|
|
|
|
| 7575859295 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859300 |
\epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{\nabla} \times \vec{E} )_k = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859302 |
\epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859304 |
\epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h}
|
|
|
|
|
| 7575859306 |
\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859308 |
\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right) = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859310 |
\hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7575859312 |
\vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
|
|
|
|
|
| 7917051060 |
\vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2}
|
|
|
|
|
| 8139187332 |
\vec{p}_{1} = \vec{p}_{2}+\vec{p}_{electron}
|
|
|
|
|
| 8257621077 |
\vec{p}_{before} = \vec{p}_{1}
|
|
|
|
|
| 8311458118 |
\vec{p}_{after} = \vec{p}_{2}+\vec{p}_{electron}
|
|
|
|
|
| 8399484849 |
\langle x^2 -2x\langle x \rangle+\langle x \rangle^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
|
|
|
|
|
| 8484544728 |
-a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x)
|
|
|
|
|
| 8485757728 |
a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(kx)
|
|
|
|
|
| 8485867742 |
\frac{2}{W} = a^2
|
|
|
|
|
| 8489593958 |
d(u v) = u dv + v du
|
|
|
|
|
| 8489593960 |
d(u v) - v du = u dv
|
|
|
|
|
| 8489593962 |
u dv = d(u v) - v du
|
|
|
|
|
| 8489593964 |
\int u dv = u v - \int v du
|
|
|
|
|
| 8494839423 |
\nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
|
|
|
|
|
| 8572657110 |
1 = \int |\psi(x)|^2 dx
|
|
|
|
|
| 8572852424 |
\vec{E} = E(\vec{r},t)
|
|
|
|
|
| 8575746378 |
\int \frac{1}{2} dx = \frac{1}{2} x
|
|
|
|
|
| 8575748999 |
\frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x)\right) = -k^2 \left(a \sin(kx) + b \cos(kx)\right)
|
|
|
|
|
| 8576785890 |
1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
|
|
|
|
|
| 8577275751 |
0 = a \sin(0) + b\cos(0)
|
|
|
|
|
| 8582885111 |
\psi(x) = a \sin(kx) + b \cos(kx)
|
|
|
|
|
| 8582954722 |
x^2 + 2xh + h^2 = (x+h)^2
|
|
|
|
|
| 8849289982 |
\psi(x)^* = a \sin(\frac{n \pi}{W} x)
|
|
|
|
|
| 8889444440 |
1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right)\right)^2 dx
|
|
|
|
|
| 9059289981 |
\psi(x) = a \sin(k x)
|
|
|
|
|
| 9285928292 |
ax^2 + bx + c = 0
|
|
|
|
|
| 9291999979 |
\vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0\vec{\nabla} \times \frac{\partial \vec{H}}{\partial t}
|
|
|
|
|
| 9294858532 |
\hat{A}^+ = \hat{A}
|
|
|
|
|
| 9385938295 |
(x+(b/(2a)))^2 = -(c/a) + (b/(2a))^2
|
|
|
|
|
| 9393939991 |
\psi(x) = -\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
|
|
|
|
|
| 9393939992 |
\psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
|
|
|
|
|
| 9394939493 |
\nabla^2 E(\vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r},t)
|
|
|
|
|
| 9429829482 |
\frac{d}{dx} y = -\sin(x) + i\cos(x)
|
|
|
|
|
| 9482113948 |
\frac{dy}{y} = i dx
|
|
|
|
|
| 9482438243 |
(\cos(x))^2 = \cos(2x)+(\sin(x))^2
|
|
|
|
|
| 9482923849 |
\exp(i x) = y
|
|
|
|
|
| 9482928242 |
\cos(2x) = (\cos(x))^2 - (\sin(x))^2
|
|
|
|
|
| 9482928243 |
\cos(2x)+(\sin(x))^2 = (\cos(x))^2
|
|
|
|
|
| 9482943948 |
\log(y) = i dx
|
|
|
|
|
| 9482948292 |
b = c
|
|
|
|
|
| 9482984922 |
\frac{d}{dx} y = (i\sin(x) + \cos(x)) i
|
|
|
|
|
| 9483928192 |
\cos(2x)+i\sin(2x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
|
|
|
|
|
| 9485384858 |
\nabla^2 E(\vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 9485747245 |
\sqrt{\frac{2}{W}} = a
|
|
|
|
|
| 9485747246 |
-\sqrt{\frac{2}{W}} = a
|
|
|
|
|
| 9492920340 |
y = \cos(x)+i \sin(x)
|
|
|
|
|
| 9494829190 |
k
|
|
|
|
|
| 9495857278 |
\psi(x=W) = 0
|
|
|
|
|
| 9499428242 |
E(\vec{r},t) = E(\vec{r})\exp(i \omega t)
|
|
|
|
|
| 9499959299 |
a + k = b + k
|
|
|
|
|
| 9582958293 |
x = \sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
|
|
|
|
|
| 9582958294 |
x+(b/(2a)) = \sqrt{(b/(2a))^2 - (c/a)}
|
|
|
|
|
| 9584821911 |
c + d = a
|
|
|
|
|
| 9585727710 |
\psi(x=0) = 0
|
|
|
|
|
| 9596004948 |
x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle
|
|
|
|
|
| 9848292229 |
dy = y i dx
|
|
|
|
|
| 9848294829 |
\frac{d}{dx} y = y i
|
|
|
|
|
| 9858028950 |
\frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
|
|
|
|
|
| 9889984281 |
2(\sin(x))^2 = 1-\cos(2x)
|
|
|
|
|
| 9919999981 |
\rho = 0
|
|
|
|
|
| 9958485859 |
\frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
|
|
|
|
|
| 9988949211 |
(\sin(x))^2 = \frac{1-\cos(2x)}{2}
|
|
|
|
|
| 9991999979 |
\vec{\nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t}
|
|
|
|
|
| 9999998870 |
\frac{\vec{p}}{\hbar} = \vec{k}
|
|
|
|
|
| 9999999870 |
\frac{p}{\hbar} = k
|
|
|
|
|
| 9999999950 |
\beta = 1/(k_{Boltzmann}T)
|
|
|
|
|
| 9999999951 |
\langle x | k \rangle = \frac{\exp(i k x)}{\sqrt{2\pi}}
|
|
|
|
|
| 9999999952 |
f(x) \delta(x-a) = f(a)
|
|
|
|
|
| 9999999953 |
\int_{-\infty}^{\infty} \delta(x) dx = 1
|
|
|
|
|
| 9999999954 |
c = 1/\sqrt{\epsilon_0 \mu_0}
|
|
|
|
|
| 9999999955 |
\vec{E} = -\vec{\nabla} \Psi
|
|
|
|
|
| 9999999956 |
\vec{F} = \frac{\partial}{\partial t}\vec{p}
|
|
|
|
|
| 9999999957 |
\vec{F} = -\vec{\nabla} V
|
|
|
|
|
| 9999999960 |
\hbar = h/(2 \pi)
|
|
|
|
|
| 9999999961 |
\frac{E}{\hbar} = \omega
|
|
|
|
|
| 9999999962 |
p = \hbar k
|
|
|
|
|
| 9999999963 |
\lambda = h/p
|
|
|
|
|
| 9999999964 |
\omega = c k
|
|
|
|
|
| 9999999965 |
E = \omega \hbar
|
|
|
|
|
| 9999999966 |
\vec{L} = \vec{r}\times\vec{p}
|
|
|
|
|
| 9999999967 |
\vec{S} = \frac{1}{\mu_0} \vec{E}\times \vec{B}
|
|
|
|
|
| 9999999968 |
x = \frac{-b-\sqrt{b^2-4ac}}{2a}
|
|
|
|
|
| 9999999969 |
x = \frac{-b+\sqrt{b^2-4ac}}{2a}
|
|
|
|
|
| 9999999970 |
\eta_1 \sin\theta_1 = \eta_2 \sin\theta_2
|
|
|
|
|
| 9999999971 |
{\cal H} = \frac{p^2}{2m}+V
|
|
|
|
|
| 9999999972 |
{\cal H} |\psi\rangle = E |\psi\rangle
|
|
|
|
|
| 9999999973 |
\left( \Delta A \right)^2 = \langle A^2 \rangle - \langle A \rangle^2
|
|
|
|
|
| 9999999974 |
\langle \psi| \hat{A} |\psi\rangle = \int_{-\infty}^{\infty} \psi^* A \psi dx
|
|
|
|
|
| 9999999975 |
\langle \psi| \hat{A} |\psi\rangle = \langle a \rangle
|
|
|
|
|
| 9999999976 |
\hat{p} = -i \hbar \frac{\partial }{\partial x}
|
|
|
|
|
| 9999999977 |
[\hat{x},\hat{p}] = i \hbar
|
|
|
|
|
| 9999999978 |
\vec{\nabla} \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial E}{\partial t}
|
|
|
|
|
| 9999999979 |
\vec{\nabla} \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}
|
|
|
|
|
| 9999999980 |
\vec{\nabla} \cdot \vec{B} = 0
|
|
|
|
|
| 9999999981 |
\vec{\nabla} \cdot \vec{E} = \rho/\epsilon_0
|
|
|
|
|
| 9999999982 |
V = I R + Q/C + L \frac{\partial I}{\partial t}
|
|
|
|
|
| 9999999983 |
C = V A/d
|
|
|
|
|
| 9999999984 |
Q = C V
|
|
|
|
|
| 9999999985 |
V = I R
|
|
|
|
|
| 9999999986 |
\left[\right]\left[\right] = \left[\right]
|
|
|
|
|
| 9999999987 |
1 atmosphere = 101.325 Pascal
|
|
|
|
|
| 9999999988 |
1 atmosphere = 14.7 pounds/(inch^2)
|
|
|
|
|
| 9999999989 |
mass_{electron} = 511000 electronVolts/(q^2)
|
|
|
|
|
| 9999999990 |
Tesla = 10000*Gauss
|
|
|
|
|
| 9999999991 |
Pascal = Newton / (meter^2)
|
|
|
|
|
| 9999999992 |
Tesla = Newton / (Ampere*meter)
|
|
|
|
|
| 9999999993 |
Farad = Coulumb / Volt
|
|
|
|
|
| 9999999995 |
Volt = Joule / Coulumb
|
|
|
|
|
| 9999999996 |
Coulomb = Ampere / second
|
|
|
|
|
| 9999999997 |
Watt = Joule / second
|
|
|
|
|
| 9999999998 |
Joule = Newton*meter
|
|
|
|
|
| 9999999999 |
Newton = kilogram*meter/(second^2)
|
|
|
|
|
| 4961662865 |
x
|
|
|
|
|
| 9110536742 |
2 x
|
|
|
|
|
| 8483686863 |
\sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x)\right)
|
|
|
|
|
| 3470587782 |
\sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
|
|
|
|
|
| 8642992037 |
2
|
|
|
|
|
| 9894826550 |
2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \left(\exp(i x)+\exp(-i x)\right)
|
|
|
|
|
| 4257214632 |
\frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)
|
|
|
|
|
| 8699789241 |
2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)
|
|
|
|
|
| 9180861128 |
2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)
|
|
|
|
|
| 2405307372 |
\sin(2 x) = 2 \sin(x) \cos(x)
|
|
|
|
|
| 3268645065 |
x
|
|
|
|
|
| 9350663581 |
\pi
|
|
|
|
|
| 8332931442 |
\exp(i \pi) = \cos(\pi)+i \sin(\pi)
|
|
|
|
|
| 6885625907 |
\exp(i \pi) = -1 + i 0
|
|
|
|
|
| 3331824625 |
\exp(i \pi) = -1
|
|
|
|
|
| 4901237716 |
1
|
|
|
|
|
| 2501591100 |
\exp(i \pi) + 1 = 0
|
|
|
|
|
| 5136652623 |
E = KE + PE
|
|
mechanical energy is the sum of the potential plus kinetic energies |
|
|
| 7915321076 |
E, KE, PE
|
|
|
|
|
| 3192374582 |
E_2, KE_2, PE_2
|
|
|
|
|
| 7875206161 |
E_2 = KE_2 + PE_2
|
|
|
|
|
| 4229092186 |
E, KE, PE
|
|
|
|
|
| 1490821948 |
E_1, KE_1, PE_1
|
|
|
|
|
| 4303372136 |
E_1 = KE_1 + PE_1
|
|
|
|
|
| 5514556106 |
E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1)
|
|
|
|
|
| 8357234146 |
KE = \frac{1}{2} m v^2
|
|
kinetic energy |
Kinetic_energy
|
|
| 3658066792 |
KE_2, v_2
|
|
|
|
|
| 8082557839 |
KE, v
|
|
|
|
|
| 7676652285 |
KE_2 = \frac{1}{2} m v_2^2
|
|
|
|
|
| 2790304597 |
KE_1, v_1
|
|
|
|
|
| 9880327189 |
KE, v
|
|
|
|
|
| 4928007622 |
KE_1 = \frac{1}{2} m v_1^2
|
|
|
|
|
| 5733146966 |
KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right)
|
|
|
|
|
| 6554292307 |
t
|
|
|
|
|
| 4270680309 |
\frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t}
|
|
|
|
|
| 5781981178 |
x^2 - y^2 = (x+y)(x-y)
|
|
difference of squares |
Difference_of_two_squares
|
|
| 5946759357 |
v_2, v_1
|
|
|
|
|
| 6675701064 |
x, y
|
|
|
|
|
| 4648451961 |
v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1)
|
|
|
|
|
| 9356924046 |
\frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t}
|
|
|
|
|
| 9397152918 |
v = \frac{v_1 + v_2}{2}
|
|
average velocity |
|
|
| 2857430695 |
a = \frac{v_2 - v_1}{t}
|
|
acceleration |
|
|
| 7735737409 |
\frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t}
|
|
|
|
|
| 4784793837 |
\frac{KE_2 - KE_1}{t} = m v a
|
|
|
|
|
| 5345738321 |
F = m a
|
|
Newton's second law of motion |
Newton%27s_laws_of_motion#Newton's_second_law
|
|
| 2186083170 |
\frac{KE_2 - KE_1}{t} = v F
|
|
|
|
|
| 6715248283 |
PE = -F x
|
|
potential energy |
Potential_energy
|
|
| 3852078149 |
PE_2, x_2
|
|
|
|
|
| 7388921188 |
PE, x
|
|
|
|
|
| 2431507955 |
PE_2 = -F x_2
|
|
|
|
|
| 3412641898 |
PE_1, x_1
|
|
|
|
|
| 9937169749 |
PE, x
|
|
|
|
|
| 4669290568 |
PE_1 = -F x_1
|
|
|
|
|
| 7734996511 |
PE_2 - PE_1 = -F ( x_2 - x_1 )
|
|
|
|
|
| 2016063530 |
t
|
|
|
|
|
| 7267155233 |
\frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right)
|
|
|
|
|
| 9337785146 |
v = \frac{x_2 - x_1}{t}
|
|
average velocity |
|
|
| 4872970974 |
\frac{PE_2 - PE_1}{t} = -F v
|
|
|
|
|
| 2081689540 |
t
|
|
|
|
|
| 2770069250 |
\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t}
|
|
|
|
|
| 3591237106 |
\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v
|
|
|
|
|
| 1772416655 |
\frac{E_2 - E_1}{t} = v F - F v
|
|
|
|
|
| 1809909100 |
\frac{E_2 - E_1}{t} = 0
|
|
|
|
|
| 5778176146 |
t
|
|
|
|
|
| 3806977900 |
E_2 - E_1 = 0
|
|
|
|
|
| 5960438249 |
E_1
|
|
|
|
|
| 8558338742 |
E_2 = E_1
|
|
|
|
|
| 8945218208 |
\theta_{Brewster} + \theta_{refracted} = 90^{\circ}
|
|
|
based on figure 34-27 on page 824 in 2001_HRW
|
|
| 9025853427 |
\theta_{Brewster}
|
|
|
|
|
| 1310571337 |
\theta_{refracted} = 90^{\circ} - \theta_{Brewster}
|
|
|
|
|
| 6450985774 |
n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 )
|
|
Law of Refraction |
eq 34-44 on page 819 in 2001_HRW
|
|
| 1779497048 |
\theta_{Brewster}, \theta_{refraction}
|
|
|
|
|
| 7322344455 |
\theta_1, \theta_2
|
|
|
|
|
| 2575937347 |
n_1 \sin( \theta_{Brewster} ) = n_2 \sin( \theta_{refraction} )
|
|
|
|
|
| 7696214507 |
n_1 \sin( \theta_{Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{Brewster} )
|
|
|
|
|
| 8588429722 |
\sin( 90^{\circ} - x ) = \cos( x )
|
|
|
|
|
| 7375348852 |
\theta_{Brewster}
|
|
|
|
|
| 1512581563 |
x
|
|
|
|
|
| 6831637424 |
\sin( 90^{\circ} - \theta_{Brewster} ) = \cos( \theta_{Brewster} )
|
|
|
|
|
| 3061811650 |
n_1 \sin( \theta_{Brewster} ) = n_2 \cos( \theta_{Brewster} )
|
|
|
|
|
| 7857757625 |
n_1
|
|
|
|
|
| 9756089533 |
\sin( \theta_{Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{Brewster} )
|
|
|
|
|
| 5632428182 |
\cos( \theta_{Brewster} )
|
|
|
|
|
| 2768857871 |
\frac{\sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )} = \frac{n_2}{n_1}
|
|
|
|
|
| 4968680693 |
\tan( x ) = \frac{ \sin( x )}{\cos( x )}
|
|
|
|
|
| 7321695558 |
\theta_{Brewster}
|
|
|
|
|
| 9906920183 |
x
|
|
|
|
|
| 4501377629 |
\tan( \theta_{Brewster} ) = \frac{ \sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )}
|
|
|
|
|
| 3417126140 |
\tan( \theta_{Brewster} ) = \frac{ n_2 }{ n_1 }
|
|
|
|
|
| 5453995431 |
\arctan{ x }
|
|
|
|
|
| 6023986360 |
x
|
|
|
|
|
| 8495187962 |
\theta_{Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }
|
|
|
|
|
| 9040079362 |
f
|
|
|
|
|
| 9565166889 |
T
|
|
|
|
|
| 5426308937 |
v = \frac{d}{t}
|
|
|
|
|
| 7476820482 |
C
|
|
|
|
|
| 1277713901 |
d
|
|
|
|
|
| 6946088325 |
v = \frac{C}{t}
|
|
|
|
|
| 6785303857 |
C = 2 \pi r
|
|
|
|
|
| 4816195267 |
C, r
|
|
|
|
|
| 2113447367 |
C_{\rm{Earth\ orbit}}, r_{\rm{Earth\ orbit}}
|
|
|
|
|
| 6348260313 |
C_{\rm{Earth\ orbit}} = 2 \pi r_{\rm{Earth\ orbit}}
|
|
|
|
|
| 3847777611 |
C, v, t
|
|
|
|
|
| 6406487188 |
C_{\rm{Earth\ orbit}}, v_{\rm{Earth\ orbit}}, t_{\rm{Earth\ orbit}}
|
|
|
|
|
| 3046191961 |
v_{\rm{Earth\ orbit}} = \frac{C_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
|
|
|
|
|
| 3080027960 |
v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
|
|
|
|
|
| 7175416299 |
t_{\rm{Earth\ orbit}} = 1 {\rm{year}}
|
|
|
|
|
| 3219318145 |
\frac{365 {\rm days}}{1 {\rm year}} \frac{24 {\rm hours}}{1 {\rm day}} \frac{60 {\rm minutes}}{1 {\rm hour}} \frac{60 {\rm seconds}}{1 {\rm minute}}
|
|
|
|
|
| 8721295221 |
t_{\rm{Earth\ orbit}} = 3.16 10^7 {\rm{seconds}}
|
|
|
|
|
| 4593428198 |
v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{3.16\ 10^7 {\rm seconds}}
|
|
|
|
|
| 3472836147 |
r_{\rm{Earth\ orbit}} = 1.496\ 10^8 {\rm km}
|
|
|
|
|
| 6998364753 |
v_{\rm{Earth\ orbit}} = \frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}}
|
|
|
|
|
| 4180845508 |
v_{\rm{Earth\ orbit}} = 29.8 \frac{{\rm km}}{{\rm sec}}
|
|
|
|
|
| 4662369843 |
x' = \gamma (x - v t)
|
|
|
|
|
| 2983053062 |
x = \gamma (x' + v t')
|
|
|
|
|
| 3426941928 |
x = \gamma ( \gamma (x - v t) + v t' )
|
|
|
|
|
| 2096918413 |
x = \gamma ( \gamma x - \gamma v t + v t' )
|
|
|
|
|
| 7741202861 |
x = \gamma^2 x - \gamma^2 v t + \gamma v t'
|
|
|
|
|
| 7337056406 |
\gamma^2 x
|
|
|
|
|
| 4139999399 |
x - \gamma^2 x = - \gamma^2 v t + \gamma v t'
|
|
|
|
|
| 3495403335 |
x
|
|
|
|
|
| 9031609275 |
x (1 - \gamma^2 ) = - \gamma^2 v t + \gamma v t'
|
|
|
|
|
| 8014566709 |
\gamma^2 v t
|
|
|
|
|
| 9409776983 |
x (1 - \gamma^2 ) + \gamma^2 v t = \gamma v t'
|
|
|
|
|
| 2226340358 |
\gamma v
|
|
|
|
|
| 6417705759 |
\frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = \gamma v t'
|
|
|
|
|
| 8730201316 |
\frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t = t'
|
|
|
first term was multiplied by \gamma/\gamma
|
|
| 1974334644 |
\frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = t'
|
|
|
|
|
| 5148266645 |
t' = \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t
|
|
|
|
|
| 4287102261 |
x^2 + y^2 + z^2 = c^2 t^2
|
|
|
describes a spherical wavefront
|
|
| 1201689765 |
x'^2 + y'^2 + z'^2 = c^2 t'^2
|
|
|
describes a spherical wavefront for an observer in a moving frame of reference
|
|
| 7057864873 |
y' = y
|
|
|
frame of reference is moving only along x direction
|
|
| 8515803375 |
z' = z
|
|
|
frame of reference is moving only along x direction
|
|
| 6153463771 |
3
|
|
|
|
|
| 4746576736 |
asdf
|
|
|
|
|
| 1027270623 |
minga
|
|
|
|
|
| 6101803533 |
ming
|
|
|
|
|
| 9231495607 |
a = b
|
|
|
|
|
| 8664264194 |
b + 2
|
|
|
|
|
| 9805063945 |
\gamma^2 (x - v t)^2 + y^2 + z^2 = c^2 \gamma^2 \left( t + \frac{ 1 - \gamma^2 }{ \gamma^2 } \frac{x}{v} \right)^2
|
|
|
|
|
| 4057686137 |
C \to C_{\rm{Earth\ orbit}}
|
|
|
|
|
| 2346150725 |
r \to r_{\rm{Earth\ orbit}}
|
|
|
|
|
| 9753878784 |
v \to v_{\rm{Earth\ orbit}}
|
|
|
|
|
| 8135396036 |
t \to t_{\rm{Earth\ orbit}}
|
|
|
|
|
| 2773628333 |
\theta_1 \to \theta_{Brewster}
|
|
|
|
|
| 7154592211 |
\theta_2 \to \theta_{\rm refracted}
|
|
|
|
|
| 3749492596 |
E \to E_1
|
|
|
|
|
| 1258245373 |
E \to E_2
|
|
|
|
|
| 4147101187 |
KE \to KE_1
|
|
|
|
|
| 3809726424 |
PE \to PE_1
|
|
|
|
|
| 6383056612 |
KE \to KE_2
|
|
|
|
|
| 5075406409 |
PE \to PE_2
|
|
|
|
|
| 5398681503 |
v \to v_1
|
|
|
|
|
| 9305761407 |
v \to v_2
|
|
|
|
|
| 4218009993 |
x \to x_1
|
|
|
|
|
| 4188639044 |
x \to x_2
|
|
|
|
|
| 8173074178 |
x \to v_2
|
|
|
|
|
| 1025759423 |
y \to v_1
|
|
|
|
|
| 1935543849 |
\gamma^2 x^2 - \gamma^2 2 x v t + \gamma^2 v^2 t^2 + y^2 + z^2 = c^2 \gamma^2 \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x^2}{\gamma^2} + c^2 \gamma^2 2 t \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x}{\gamma} + c^2 \gamma^2 t^2
|
|
|
|
|
| 1586866563 |
\left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right) = t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)
|
|
|
|
|
| 3182633789 |
\gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1
|
|
|
|
|
| 1916173354 |
-\gamma^2 v^2 + c^2 \gamma^2 = c^2
|
|
|
|
|
| 2076171250 |
-\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} = 0
|
|
|
|
|
| 5284610349 |
\gamma^2
|
|
|
|
|
| 2417941373 |
- c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 - \gamma^2
|
|
|
|
|
| 5787469164 |
1 - \gamma^2
|
|
|
|
|
| 1639827492 |
- c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2} = 1
|
|
|
|
|
| 5669500954 |
v^2 \gamma^2
|
|
|
|
|
| 5763749235 |
-c^2 + c^2 \gamma^2 = v^2 \gamma^2
|
|
|
|
|
| 6408214498 |
c^2
|
|
|
|
|
| 2999795755 |
c^2 \gamma^2 = v^2 \gamma^2 + c^2
|
|
|
|
|
| 3412946408 |
v^2 \gamma^2
|
|
|
|
|
| 2542420160 |
c^2 \gamma^2 - v^2 \gamma^2 = c^2
|
|
|
|
|
| 7743841045 |
\gamma^2
|
|
|
|
|
| 7513513483 |
\gamma^2 (c^2 - v^2) = c^2
|
|
|
|
|
| 8571466509 |
c^2 - \gamma^2
|
|
|
|
|
| 7906112355 |
\gamma^2 = \frac{c^2}{c^2 - \gamma^2}
|
|
|
|
|
| 3060139639 |
1/2
|
|
|
|
|
| 1528310784 |
\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
|
|
|
|
|
| 8360117126 |
\gamma = \frac{-1}{\sqrt{1-\frac{v^2}{c^2}}}
|
|
|
not a physically valid result in this context
|
|
| 3366703541 |
a = \frac{v - v_0}{t}
|
|
|
acceleration is the average change in speed over a duration
|
|
| 7083390553 |
t
|
|
|
|
|
| 4748157455 |
a t = v - v_0
|
|
|
|
|
| 6417359412 |
v_0
|
|
|
|
|
| 4798787814 |
a t + v_0 = v
|
|
|
|
|
| 3462972452 |
v = v_0 + a t
|
|
|
|
|
| 3411994811 |
v_{\rm ave} = \frac{d}{t}
|
|
|
|
|
| 6175547907 |
v_{\rm ave} = \frac{v + v_0}{2}
|
|
|
|
|
| 9897284307 |
\frac{d}{t} = \frac{v + v_0}{2}
|
|
|
|
|
| 8865085668 |
t
|
|
|
|
|
| 8706092970 |
d = \left(\frac{v + v_0}{2}\right)t
|
|
|
|
|
| 7011114072 |
d = \frac{(v_0 + a t) + v_0}{2} t
|
|
|
|
|
| 1265150401 |
d = \frac{2 v_0 + a t}{2} t
|
|
|
|
|
| 9658195023 |
d = v_0 t + \frac{1}{2} a t^2
|
|
|
|
|
| 5799753649 |
2
|
|
|
|
|
| 7215099603 |
v^2 = v_0^2 + 2 a t v_0 + a^2 t^2
|
|
|
|
|
| 5144263777 |
v^2 = v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right)
|
|
|
|
|
| 7939765107 |
v^2 = v_0^2 + 2 a d
|
|
|
|
|
| 6729698807 |
v_0
|
|
|
|
|
| 9759901995 |
v - v_0 = a t
|
|
|
|
|
| 2242144313 |
a
|
|
|
|
|
| 1967582749 |
t = \frac{v - v_0}{a}
|
|
|
|
|
| 5733721198 |
d = \frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right)
|
|
|
|
|
| 5611024898 |
d = \frac{1}{2 a} (v^2 - v_0^2)
|
|
|
|
|
| 5542390646 |
2 a
|
|
|
|
|
| 8269198922 |
2 a d = v^2 - v_0^2
|
|
|
|
|
| 9070454719 |
v_0^2
|
|
|
|
|
| 4948763856 |
2 a d + v_0^2 = v^2
|
|
|
|
|
| 9645178657 |
a t
|
|
|
|
|
| 6457044853 |
v - a t = v_0
|
|
|
|
|
| 1259826355 |
d = (v - a t) t + \frac{1}{2} a t^2
|
|
|
|
|
| 4580545876 |
d = v t - a t^2 + \frac{1}{2} a t^2
|
|
|
|
|
| 6421241247 |
d = v t - \frac{1}{2} a t^2
|
|
|
|
|
| 4182362050 |
Z = |Z| \exp( i \theta )
|
|
|
Z \in \mathbb{C}
|
|
| 1928085940 |
Z^* = |Z| \exp( -i \theta )
|
|
|
|
|
| 9191880568 |
Z Z^* = |Z| |Z| \exp( -i \theta ) \exp( i \theta )
|
|
|
|
|
| 3350830826 |
Z Z^* = |Z|^2
|
|
|
|
|
| 8396997949 |
I = | A + B |^2
|
|
|
intensity of two waves traveling opposite directions on same path
|
|
| 4437214608 |
Z
|
|
|
|
|
| 5623794884 |
A + B
|
|
|
|
|
| 2236639474 |
(A + B)(A + B)^* = |A + B|^2
|
|
|
|
|
| 1020854560 |
I = (A + B)(A + B)^*
|
|
|
|
|
| 6306552185 |
I = (A + B)(A^* + B^*)
|
|
|
|
|
| 8065128065 |
I = A A^* + B B^* + A B^* + B A^*
|
|
|
|
|
| 9761485403 |
Z
|
|
|
|
|
| 8710504862 |
A
|
|
|
|
|
| 4075539836 |
A A^* = |A|^2
|
|
|
|
|
| 6529120965 |
B
|
|
|
|
|
| 1511199318 |
Z
|
|
|
|
|
| 7107090465 |
B B^* = |B|^2
|
|
|
|
|
| 5125940051 |
I = |A|^2 + B B^* + A B^* + B A^*
|
|
|
|
|
| 1525861537 |
I = |A|^2 + |B|^2 + A B^* + B A^*
|
|
|
|
|
| 1742775076 |
Z \to B
|
|
|
|
|
| 7607271250 |
\theta \to \phi
|
|
|
|
|
| 4192519596 |
B = |B| \exp(i \phi)
|
|
|
|
|
| 2064205392 |
A
|
|
|
|
|
| 1894894315 |
Z
|
|
|
|
|
| 1357848476 |
A = |A| \exp(i \theta)
|
|
|
|
|
| 6555185548 |
A^* = |A| \exp(-i \theta)
|
|
|
|
|
| 4504256452 |
B^* = |B| \exp(-i \phi)
|
|
|
|
|
| 7621705408 |
I = |A|^2 + |B|^2 + |A| |B| \exp(-i \theta) \exp(i \phi) + |A| |B| \exp(i \theta) \exp(-i \phi)
|
|
|
|
|
| 3085575328 |
I = |A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi))
|
|
|
|
|
| 4935235303 |
x
|
|
|
|
|
| 2293352649 |
\theta - \phi
|
|
|
|
|
| 3660957533 |
\cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
|
|
|
|
|
| 3967985562 |
2
|
|
|
|
|
| 2700934933 |
2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
|
|
|
|
|
| 8497631728 |
I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )
|
|
|
|
|
| 6774684564 |
\theta = \phi
|
|
|
for coherent waves
|
|
| 2099546947 |
I = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )
|
|
|
|
|
| 2719691582 |
|A| = |B|
|
|
|
in a loop
|
|
| 3257276368 |
I = |A|^2 + |A|^2 + |A| |A| 2
|
|
|
|
|
| 8283354808 |
I_{\rm coherent} = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )
|
|
|
|
|
| 8046208134 |
I_{\rm coherent} = |A|^2 + |A|^2 + |A| |A| 2
|
|
|
|
|
| 1172039918 |
I_{\rm coherent} = 4 |A|^2
|
|
|
|
|
| 8602221482 |
\langle \cos(\theta - \phi) \rangle = 0
|
|
|
incoherent light source
|
|
| 6240206408 |
I_{\rm incoherent} = |A|^2 + |B|^2
|
|
|
|
|
| 6529793063 |
I_{\rm incoherent} = |A|^2 + |A|^2
|
|
|
|
|
| 3060393541 |
I_{\rm incoherent} = 2|A|^2
|
|
|
|
|
| 6556875579 |
\frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2
|
|
|
|
|
| 8750500372 |
f
|
|
|
|
|
| 7543102525 |
g
|
|
|
|
|
| 8267133829 |
a = b
|
|
|
|
|
| 7968822469 |
c = d
|
|
|
|
|
| 3169580383 |
\vec{a} = \frac{d\vec{v}}{dt}
|
|
|
acceleration is the change in speed over a duration
|
|
| 8602512487 |
\vec{a} = a_x \hat{x} + a_y \hat{y}
|
|
|
decompose acceleration into two components
|
|
| 4158986868 |
a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt}
|
|
|
|
|
| 5349866551 |
\vec{v} = v_x \hat{x} + v_y \hat{y}
|
|
|
|
|
| 7729413831 |
a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right)
|
|
|
|
|
| 1819663717 |
a_x = \frac{d}{dt} v_x
|
|
|
|
|
| 8228733125 |
a_y = \frac{d}{dt} v_y
|
|
|
|
|
| 9707028061 |
a_x = 0
|
|
|
|
|
| 2741489181 |
a_y = -g
|
|
|
|
|
| 3270039798 |
2
|
|
|
|
|
| 8880467139 |
2
|
|
|
|
|
| 8750379055 |
0 = \frac{d}{dt} v_x
|
|
|
|
|
| 1977955751 |
-g = \frac{d}{dt} v_y
|
|
|
|
|
| 6672141531 |
dt
|
|
|
|
|
| 1702349646 |
-g dt = d v_y
|
|
|
|
|
| 8584698994 |
-g \int dt = \int d v_y
|
|
|
|
|
| 9973952056 |
-g t = v_y - v_{0, y}
|
|
|
|
|
| 4167526462 |
v_{0, y}
|
|
|
|
|
| 6572039835 |
-g t + v_{0, y} = v_y
|
|
|
|
|
| 7252338326 |
v_y = \frac{dy}{dt}
|
|
|
|
|
| 6204539227 |
-g t + v_{0, y} = \frac{dy}{dt}
|
|
|
|
|
| 1614343171 |
dt
|
|
|
|
|
| 8145337879 |
-g t dt + v_{0, y} dt = dy
|
|
|
|
|
| 8808860551 |
-g \int t dt + v_{0, y} \int dt = \int dy
|
|
|
|
|
| 2858549874 |
- \frac{1}{2} g t^2 + v_{0, y} t = y - y_0
|
|
|
|
|
| 6098638221 |
y_0
|
|
|
|
|
| 2461349007 |
- \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y
|
|
|
|
|
| 8717193282 |
dt
|
|
|
|
|
| 1166310428 |
0 dt = d v_x
|
|
|
|
|
| 2366691988 |
\int 0 dt = \int d v_x
|
|
|
|
|
| 1676472948 |
0 = v_x - v_{0, x}
|
|
|
|
|
| 1439089569 |
v_{0, x}
|
|
|
|
|
| 6134836751 |
v_{0, x} = v_x
|
|
|
|
|
| 8460820419 |
v_x = \frac{dx}{dt}
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| 7455581657 |
v_{0, x} = \frac{dx}{dt}
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| 8607458157 |
dt
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| 1963253044 |
v_{0, x} dt = dx
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| 3676159007 |
v_{0, x} \int dt = \int dx
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| 9882526611 |
v_{0, x} t = x - x_0
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| 3182907803 |
x_0
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| 8486706976 |
v_{0, x} t + x_0 = x
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| 1306360899 |
x = v_{0, x} t + x_0
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| 7049769409 |
2
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| 9341391925 |
\vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y}
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| 6410818363 |
\theta
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| 7391837535 |
\cos(\theta) = \frac{v_{0, x}}{v_0}
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| 7376526845 |
\sin(\theta) = \frac{v_{0, y}}{v_0}
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| 5868731041 |
v_0
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| 6083821265 |
v_0 \cos(\theta) = v_{0, x}
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| 5438722682 |
x = v_0 t \cos(\theta) + x_0
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| 5620558729 |
v_0
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| 8949329361 |
v_0 \sin(\theta) = v_{0, y}
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| 1405465835 |
y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0
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| 9862900242 |
y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0
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| 6050070428 |
v_{0, x}
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| 3274926090 |
t = \frac{x - x_0}{v_{0, x}}
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| 7354529102 |
y = - \frac{1}{2} g \left( \frac{x - x_0}{v_{0, x}} \right)^2 + v_{0, y} \frac{x - x_0}{v_{0, x}} + y_0
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| 8406170337 |
y \to y_f
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| 2403773761 |
t \to t_f
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| 5379546684 |
y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
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| 5373931751 |
t = t_f
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| 9112191201 |
y_f = 0
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| 8198310977 |
0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
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| 1650441634 |
y_0 = 0
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define coordinate system such that initial height is at origin
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| 1087417579 |
0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)
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| 4829590294 |
t_f
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| 2086924031 |
0 = - \frac{1}{2} g t_f + v_0 \sin(\theta)
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| 6974054946 |
\frac{1}{2} g t_f
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| 1191796961 |
\frac{1}{2} g t_f = v_0 \sin(\theta)
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| 2510804451 |
2/g
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| 4778077984 |
t_f = \frac{2 v_0 \sin(\theta)}{g}
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| 3273630811 |
x \to x_f
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| 6732786762 |
t \to t_f
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| 3485125659 |
x_f = v_0 t_f \cos(\theta) + x_0
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| 4370074654 |
t = t_f
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| 2378095808 |
x_f = x_0 + d
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| 4268085801 |
x_0 + d = v_0 t_f \cos(\theta) + x_0
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| 8072682558 |
x_0
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| 7233558441 |
d = v_0 t_f \cos(\theta)
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| 2297105551 |
d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)
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| 7587034465 |
\theta
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| 7214442790 |
x
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| 2519058903 |
\sin(2 \theta) = 2 \sin(\theta) \cos(\theta)
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| 8922441655 |
d = \frac{v_0^2}{g} \sin(2 \theta)
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| 5667870149 |
\theta
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| 1541916015 |
\theta = \frac{\pi}{4}
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| 3607070319 |
d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)
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| 5353282496 |
d = \frac{v_0^2}{g}
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